{"id":6178,"date":"2023-04-18T16:43:20","date_gmt":"2023-04-18T07:43:20","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=6178"},"modified":"2024-07-30T10:09:04","modified_gmt":"2024-07-30T01:09:04","slug":"sympy-%e3%81%a7%e5%b8%b8%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%bc%8f","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/sympy-%e3%81%a7%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/sympy-%e3%81%a7%e5%b8%b8%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%bc%8f\/","title":{"rendered":"SymPy \u3067\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f"},"content":{"rendered":"<p><!--more--><\/p>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u5fc5\u8981\u306a\u30d1\u30c3\u30b1\u30fc\u30b8\u306e-import-\u3068\u8a2d\u5b9a\">\u5fc5\u8981\u306a\u30d1\u30c3\u30b1\u30fc\u30b8\u306e import \u3068\u8a2d\u5b9a<\/h3>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[1]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># 1\u6587\u5b57\u5909\u6570\u306e Symbol \u306e\u5b9a\u7fa9\u304c\u7701\u7565\u3067\u304d\u308b<\/span>\r\n<span class=\"kn\">from<\/span> <span class=\"nn\">sympy.abc<\/span> <span class=\"kn\">import<\/span> <span class=\"o\">*<\/span>\r\n<span class=\"c1\"># a Python library for symbolic mathematics<\/span>\r\n<span class=\"kn\">from<\/span> <span class=\"nn\">sympy<\/span> <span class=\"kn\">import<\/span> <span class=\"o\">*<\/span>\r\n\r\n<span class=\"c1\"># SymPy Plotting Backends (SPB): \u30b0\u30e9\u30d5\u3092\u63cf\u304f\u969b\u306b\u5229\u7528<\/span>\r\n<span class=\"kn\">from<\/span> <span class=\"nn\">spb<\/span> <span class=\"kn\">import<\/span> <span class=\"o\">*<\/span>\r\n\r\n<span class=\"c1\"># \u30b0\u30e9\u30d5\u3092 SVG \u3067 Notebook \u306b\u30a4\u30f3\u30e9\u30a4\u30f3\u8868\u793a<\/span>\r\n<span class=\"o\">%<\/span><span class=\"k\">config<\/span> InlineBackend.figure_formats = ['svg']\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u5909\u6570\u5206\u96e2\u6cd5\">\u5909\u6570\u5206\u96e2\u6cd5<\/h3>\n<h4 id=\"\u4f8b\u984c\">\u4f8b\u984c<\/h4>\n<p>$$ \\frac{dy}{dx} = -2 x\\, y $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<ul>\n<li><code>y<\/code> \u3092\uff08\u5909\u6570\u3067\u306f\u306a\u304f\uff09\u300c\u95a2\u6570\u300d\u3068\u3057\u3066\u5b9a\u7fa9\u3059\u308b\u306b\u306f <code>y = Function('y')<\/code><\/li>\n<li>\u65b9\u7a0b\u5f0f\u30fb\u7b49\u5f0f\u306f <code>Eq(\u5de6\u8fba, \u53f3\u8fba)<\/code><\/li>\n<li>\u5fae\u5206\u3067\u3042\u308b\u3053\u3068\u3092\u8868\u3059\u306b\u306f <code>Derivative()<\/code>\uff08<code>diff(y(x), x)<\/code> \u3060\u3068\u5b9f\u969b\u306b\u5fae\u5206\u3057\u3066\u3057\u307e\u3046\uff09<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[2]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># y \u3092 x \u306e\u95a2\u6570\u3068\u3057\u3066\u5ba3\u8a00<\/span>\r\n<span class=\"n\">y<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Function<\/span><span class=\"p\">(<\/span><span class=\"s1\">'y'<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"c1\"># \u89e3\u304f\u3079\u304d\u5fae\u5206\u65b9\u7a0b\u5f0f<\/span>\r\n<span class=\"n\">eq<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">Derivative<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"o\">-<\/span><span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"o\">*<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">eq<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[2]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{d}{d x} y{\\left(x \\right)} = -2 x y{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>SymPy \u3067\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u306b\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b <code>dsolve()<\/code> \u95a2\u6570\u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[3]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eq<\/span><span class=\"p\">,<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[3]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = C_{1} e^{-x^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u4e0a\u306e\u51fa\u529b\u3067 $C_1$ \u306f\u7a4d\u5206\u5b9a\u6570\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u3061\u306a\u307f\u306b\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\uff0c\u89e3\u304f\u3079\u304d\u65b9\u7a0b\u5f0f\u3092\u53f3\u8fba\u304c\u30bc\u30ed\u306b\u306a\u308b\u3088\u3046\u306b<\/p>\n<p>$$\\frac{dy}{dx} + 2 x\\, y =0$$<\/p>\n<p>\u3068\u3057\u3066\uff0c\u5de6\u8fba\u306e\u307f\u3092 <code>dsolve()<\/code> \u306e\u5f15\u6570\u306b\u3057\u3066\u3082\uff0c\u540c\u3058\u7b54\u3048\u304c\u3067\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[4]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">Derivative<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">+<\/span><span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"o\">*<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[4]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = C_{1} e^{-x^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u30de\u30eb\u30b5\u30b9\u306e\u4eba\u53e3\u30e2\u30c7\u30eb\">\u30de\u30eb\u30b5\u30b9\u306e\u4eba\u53e3\u30e2\u30c7\u30eb<\/h3>\n<p>$$\\frac{dN}{dt} = \\gamma \\, N$$<\/p>\n<p>\u3092\u521d\u671f\u6761\u4ef6 $t = t_0$ \u3067 $N(t_0) = N_0$ \u3068\u3057\u3066\u89e3\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[5]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u95a2\u6570\u304a\u3088\u3073\u5909\u6570\u306e\u5ba3\u8a00<\/span>\r\n<span class=\"n\">N<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Function<\/span><span class=\"p\">(<\/span><span class=\"s1\">'N'<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'gamma'<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">eqm<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">Derivative<\/span><span class=\"p\">(<\/span><span class=\"n\">N<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">),<\/span> <span class=\"n\">t<\/span><span class=\"p\">),<\/span> <span class=\"n\">gamma<\/span><span class=\"o\">*<\/span><span class=\"n\">N<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">eqm<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[5]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{d}{d t} N{\\left(t \\right)} = \\gamma N{\\left(t \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[6]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u521d\u671f\u5024\u5909\u6570\u306e\u5ba3\u8a00<\/span>\r\n<span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'t0, N0'<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"c1\"># \u521d\u671f\u6761\u4ef6\u306f ics={} \u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b...<\/span>\r\n<span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eqm<\/span><span class=\"p\">,<\/span> <span class=\"n\">N<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">),<\/span> <span class=\"n\">ics<\/span><span class=\"o\">=<\/span><span class=\"p\">{<\/span><span class=\"n\">N<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">subs<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">,<\/span><span class=\"n\">t0<\/span><span class=\"p\">):<\/span><span class=\"n\">N0<\/span><span class=\"p\">})<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[6]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle N{\\left(t \\right)} = N_{0} e^{\\gamma t} e^{-\\gamma t_{0}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u53c2\u8003\uff1a\u7c73\u56fd\u306e\u4eba\u53e3\u3068\u30de\u30eb\u30b5\u30b9\u30e2\u30c7\u30eb\">\u53c2\u8003\uff1a\u7c73\u56fd\u306e\u4eba\u53e3\u3068\u30de\u30eb\u30b5\u30b9\u30e2\u30c7\u30eb<\/h4>\n<ul>\n<li>\u53c2\u8003\uff1a<a href=\"https:\/\/www.nippyo.co.jp\/shop\/book\/1240.html\">\u300c\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u6570\u5b66\u30e2\u30c7\u30eb\u3092\u4f5c\u308d\u3046\u300d \u30d0\u30fc\u30b8\u30a7\u30b9\uff0c\u30dc\u30ea\u30fc\u8457\uff0c\u65e5\u672c\u8a55\u8ad6\u793e<\/a><\/li>\n<\/ul>\n<p>\u4e0a\u8a18\u306b\u3088\u308c\u3070\uff0c1790\u5e74\u304b\u30891930\u5e74\u306e\u30a2\u30e1\u30ea\u30ab\u306e\u4eba\u53e3\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u3063\u3066\u3044\u308b\uff08\u4eba\u53e3\u306e\u5358\u4f4d\u306f\u767e\u4e07\u4eba\uff09\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[7]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'usa'<\/span><span class=\"p\">)<\/span>\r\n<span class=\"c1\"># \u884c\u5217 Matrix \u3092\u4f7f\u3063\u3066\u307f\u308b\u3002<\/span>\r\n<span class=\"n\">usa<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Matrix<\/span><span class=\"p\">([<\/span>\r\n<span class=\"c1\"># \u897f\u66a6, \u4eba\u53e3\uff08\u767e\u4e07\u4eba\uff09<\/span>\r\n  <span class=\"p\">[<\/span><span class=\"mi\">1790<\/span><span class=\"p\">,<\/span>  <span class=\"mf\">3.9<\/span><span class=\"p\">],<\/span> \r\n  <span class=\"p\">[<\/span><span class=\"mi\">1800<\/span><span class=\"p\">,<\/span>  <span class=\"mf\">5.3<\/span><span class=\"p\">],<\/span>\r\n  <span class=\"p\">[<\/span><span class=\"mi\">1810<\/span><span class=\"p\">,<\/span>  <span class=\"mf\">7.2<\/span><span class=\"p\">],<\/span>\r\n  <span class=\"p\">[<\/span><span class=\"mi\">1820<\/span><span class=\"p\">,<\/span>  <span class=\"mf\">9.6<\/span><span class=\"p\">],<\/span>\r\n  <span class=\"p\">[<\/span><span class=\"mi\">1830<\/span><span class=\"p\">,<\/span> <span class=\"mf\">12.9<\/span><span class=\"p\">],<\/span>\r\n  <span class=\"p\">[<\/span><span class=\"mi\">1840<\/span><span class=\"p\">,<\/span> <span class=\"mf\">17.1<\/span><span class=\"p\">],<\/span>\r\n  <span class=\"p\">[<\/span><span class=\"mi\">1850<\/span><span class=\"p\">,<\/span> <span class=\"mf\">23.2<\/span><span class=\"p\">],<\/span>\r\n  <span class=\"p\">[<\/span><span class=\"mi\">1860<\/span><span class=\"p\">,<\/span> <span class=\"mf\">31.4<\/span><span class=\"p\">],<\/span>\r\n  <span class=\"p\">[<\/span><span class=\"mi\">1870<\/span><span class=\"p\">,<\/span> <span class=\"mf\">38.6<\/span><span class=\"p\">],<\/span>\r\n  <span class=\"p\">[<\/span><span class=\"mi\">1880<\/span><span class=\"p\">,<\/span> <span class=\"mf\">50.2<\/span><span class=\"p\">],<\/span>\r\n  <span class=\"p\">[<\/span><span class=\"mi\">1890<\/span><span class=\"p\">,<\/span> <span class=\"mf\">62.9<\/span><span class=\"p\">],<\/span>\r\n  <span class=\"p\">[<\/span><span class=\"mi\">1900<\/span><span class=\"p\">,<\/span> <span class=\"mf\">76.0<\/span><span class=\"p\">],<\/span>\r\n  <span class=\"p\">[<\/span><span class=\"mi\">1910<\/span><span class=\"p\">,<\/span> <span class=\"mf\">92.0<\/span><span class=\"p\">],<\/span>\r\n  <span class=\"p\">[<\/span><span class=\"mi\">1920<\/span><span class=\"p\">,<\/span><span class=\"mf\">106.5<\/span><span class=\"p\">],<\/span>\r\n  <span class=\"p\">[<\/span><span class=\"mi\">1930<\/span><span class=\"p\">,<\/span><span class=\"mf\">123.2<\/span><span class=\"p\">]<\/span>\r\n<span class=\"p\">])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$t_0 = 1790$ \uff08\u5e74\uff09\u3068\u3059\u308b\u3068\uff0c$N_0 = 3.9$ \u3002<\/p>\n<p>Python \u306e\u8981\u7d20\u3092\u53d6\u308a\u51fa\u3059\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\u306f <code>0<\/code> \u30bc\u30ed\u306f\u3058\u307e\u308a\u3067\u3042\u308b\u304b\u3089&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[8]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">t0<\/span> <span class=\"o\">=<\/span> <span class=\"n\">usa<\/span><span class=\"p\">[<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">]<\/span> <span class=\"c1\"># 1\u884c\u76ee\u306e1\u5217\u76ee<\/span>\r\n<span class=\"n\">N0<\/span> <span class=\"o\">=<\/span> <span class=\"n\">usa<\/span><span class=\"p\">[<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">]<\/span> <span class=\"c1\"># 1\u884c\u76ee\u306e2\u5217\u76ee<\/span>\r\n<span class=\"n\">t0<\/span><span class=\"p\">,<\/span> <span class=\"n\">N0<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[8]:<\/div>\n<div class=\"output_text output_subarea output_execute_result\">\n<pre>(1790, 3.90000000000000)<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u6b8b\u308a\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc $\\gamma$ \u306f\uff0c\u5225\u306e\u6642\u523b $t_1$ \u306b\u304a\u3051\u308b $N(t_1)$ \u304b\u3089\u6c42\u3081\u3066\u307f\u308b\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\nN(t_1) &amp;=&amp; N_0 e^{\\gamma\\, (t_1 -t_0)} \\\\<br \/>\n\\log \\frac{N(t_1)}{N_0} &amp;=&amp; \\gamma\\, (t_1 -t_0) \\\\<br \/>\n\\therefore\\ \\ \\gamma &amp;=&amp; \\frac{1}{t_1 -t_0} \\log \\frac{N(t_1)}{N_0}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u305f\u3068\u3048\u3070\uff0c$t_1 = 1830$ \u3068\u3059\u308b\u3068\uff08Python \u306e\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\u306f <code>0<\/code> \u30bc\u30ed\u306f\u3058\u307e\u308a\u3060\u304b\u3089\uff09&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[9]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">t1<\/span> <span class=\"o\">=<\/span> <span class=\"n\">usa<\/span><span class=\"p\">[<\/span><span class=\"mi\">4<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">]<\/span>  <span class=\"c1\"># 5\u884c\u76ee\u306e1\u5217\u76ee <\/span>\r\n<span class=\"n\">N1<\/span> <span class=\"o\">=<\/span> <span class=\"n\">usa<\/span><span class=\"p\">[<\/span><span class=\"mi\">4<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">]<\/span>  <span class=\"c1\"># 5\u884c\u76ee\u306e2\u5217\u76ee <\/span>\r\n\r\n<span class=\"mi\">1<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"n\">t1<\/span><span class=\"o\">-<\/span><span class=\"n\">t0<\/span><span class=\"p\">)<\/span> <span class=\"o\">*<\/span> <span class=\"n\">log<\/span><span class=\"p\">(<\/span><span class=\"n\">N1<\/span><span class=\"o\">\/<\/span><span class=\"n\">N0<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[9]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle 0.0299062689558006$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u3057\u305f\u304c\u3063\u3066\uff0c\u30de\u30eb\u30b5\u30b9\u306e\u4eba\u53e3\u30e2\u30c7\u30eb\u3092 $N_m(t)$ \u3068\u3059\u308b\u3068<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[10]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"k\">def<\/span> <span class=\"nf\">Nm<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"n\">t0<\/span> <span class=\"o\">=<\/span> <span class=\"n\">usa<\/span><span class=\"p\">[<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">]<\/span>\r\n    <span class=\"n\">N0<\/span> <span class=\"o\">=<\/span> <span class=\"n\">usa<\/span><span class=\"p\">[<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">]<\/span>\r\n    <span class=\"n\">gamma<\/span> <span class=\"o\">=<\/span> <span class=\"mi\">1<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"n\">t1<\/span><span class=\"o\">-<\/span><span class=\"n\">t0<\/span><span class=\"p\">)<\/span> <span class=\"o\">*<\/span> <span class=\"n\">log<\/span><span class=\"p\">(<\/span><span class=\"n\">N1<\/span><span class=\"o\">\/<\/span><span class=\"n\">N0<\/span><span class=\"p\">)<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"n\">N0<\/span> <span class=\"o\">*<\/span> <span class=\"n\">exp<\/span><span class=\"p\">(<\/span><span class=\"n\">gamma<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"o\">-<\/span><span class=\"n\">t0<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u3053\u306e\u3088\u3046\u306b\u3057\u3066\u6c42\u3081\u3089\u308c\u305f $N_m(t)$ \u3092\uff0c\u4eba\u53e3\u30c7\u30fc\u30bf <code>usa<\/code> \u3068\u5171\u30b0\u30e9\u30d5\u306b\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[11]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># SymPy Plotting Backends (SPB) \u3067\u967d\u95a2\u6570\u3092\u63cf\u304f<\/span>\r\n\r\n<span class=\"n\">p1<\/span> <span class=\"o\">=<\/span> <span class=\"n\">plot<\/span><span class=\"p\">(<\/span><span class=\"n\">Nm<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">),<\/span> <span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1790<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1940<\/span><span class=\"p\">),<\/span> <span class=\"s2\">\"\u30de\u30eb\u30b5\u30b9\u30e2\u30c7\u30eb\"<\/span><span class=\"p\">,<\/span> \r\n          <span class=\"n\">legend<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">True<\/span><span class=\"p\">,<\/span> \r\n          <span class=\"n\">xlabel<\/span> <span class=\"o\">=<\/span> <span class=\"s2\">\"\u5e74\"<\/span><span class=\"p\">,<\/span> <span class=\"n\">ylabel<\/span> <span class=\"o\">=<\/span> <span class=\"s2\">\"\u4eba\u53e3\uff08\u5358\u4f4d\uff1a\u767e\u4e07\u4eba\uff09\"<\/span><span class=\"p\">);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-9268\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/PMathC101.svg\" alt=\"\" width=\"640\" height=\"427\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>SPB \u3067\u70b9\uff0c$x$ \u5ea7\u6a19 $y$ \u5ea7\u6a19\u306e\u6570\u5024\u30c7\u30fc\u30bf\u3092\u30b0\u30e9\u30d5\u306b\u3059\u308b\u969b\u306e\u66f8\u5f0f\u306f<\/p>\n<div class=\"highlight\">\n<pre><span class=\"n\">plot_list<\/span><span class=\"p\">([<\/span><span class=\"n\">x1<\/span><span class=\"p\">,<\/span> <span class=\"o\">...<\/span><span class=\"p\">,<\/span> <span class=\"n\">xn<\/span><span class=\"p\">],<\/span> <span class=\"p\">[<\/span><span class=\"n\">y1<\/span><span class=\"p\">,<\/span> <span class=\"o\">...<\/span><span class=\"p\">,<\/span> <span class=\"n\">yn<\/span><span class=\"p\">])<\/span>\r\n<\/pre>\n<\/div>\n<p>\u3067\u3059\u306e\u3067\uff0c<\/p>\n<ul>\n<li>\u884c\u5217 <code>usa<\/code> \u306e 1 \u5217\u76ee <code>.col(0)<\/code> \u3092\u30ea\u30b9\u30c8\u306b\u3057\u305f <code>list(usa.col(0))<\/code> \u3092$x$ \u5ea7\u6a19\uff0c<\/li>\n<li>2 \u5217\u76ee <code>list(usa.col(1))<\/code> \u3092$y$ \u5ea7\u6a19,<\/li>\n<li>\u6570\u5024\u30c7\u30fc\u30bf\u3092\u70b9 <code>is_poinst = True<\/code><\/li>\n<\/ul>\n<p>\u3067\u30b0\u30e9\u30d5\u306b\u3057\u307e\u3059\u3002<code>.col()<\/code> \u3092\u4f7f\u3044\u305f\u3044\u304c\u305f\u3081\u306b\uff0c<code>usa<\/code> \u3092\u884c\u5217\u3068\u3057\u305f\u306e\u3067\u3057\u305f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[12]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">p2<\/span> <span class=\"o\">=<\/span> <span class=\"n\">plot_list<\/span><span class=\"p\">(<\/span><span class=\"nb\">list<\/span><span class=\"p\">(<\/span><span class=\"n\">usa<\/span><span class=\"o\">.<\/span><span class=\"n\">col<\/span><span class=\"p\">(<\/span><span class=\"mi\">0<\/span><span class=\"p\">)),<\/span> <span class=\"nb\">list<\/span><span class=\"p\">(<\/span><span class=\"n\">usa<\/span><span class=\"o\">.<\/span><span class=\"n\">col<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"p\">)),<\/span> \r\n               <span class=\"s2\">\"\u30a2\u30e1\u30ea\u30ab\u306e\u4eba\u53e3\"<\/span><span class=\"p\">,<\/span> \r\n               <span class=\"c1\"># \u7dda\u3067\u7e4b\u304c\u3059\u70b9\u3067, \u51e1\u4f8b\u3092\u8868\u793a<\/span>\r\n               <span class=\"n\">is_point<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">True<\/span><span class=\"p\">,<\/span> <span class=\"n\">legend<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">True<\/span><span class=\"p\">,<\/span> \r\n               <span class=\"n\">xlabel<\/span> <span class=\"o\">=<\/span> <span class=\"s2\">\"\u5e74\"<\/span><span class=\"p\">,<\/span> <span class=\"n\">ylabel<\/span> <span class=\"o\">=<\/span> <span class=\"s2\">\"\u4eba\u53e3\uff08\u5358\u4f4d\uff1a\u767e\u4e07\u4eba\uff09\"<\/span><span class=\"p\">);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-9269\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/PMathC102.svg\" alt=\"\" width=\"640\" height=\"427\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>SPB \u3067\u4e0a\u8a182\u3064\u306e\u30b0\u30e9\u30d5\u3092\u91cd\u306d\u3066\u8868\u793a\u3055\u305b\u308b\u4f8b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[13]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">p3<\/span> <span class=\"o\">=<\/span> <span class=\"n\">p1<\/span> <span class=\"o\">+<\/span> <span class=\"n\">p2<\/span>\r\n<span class=\"n\">p3<\/span><span class=\"o\">.<\/span><span class=\"n\">show<\/span><span class=\"p\">();<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-9270\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/PMathC103.svg\" alt=\"\" width=\"640\" height=\"427\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u30de\u30eb\u30b5\u30b9\u30fb\u30e2\u30c7\u30eb\u306f $\\gamma &gt; 0$ \u306e\u5834\u5408\u306b\u306f\u4eba\u53e3\u306e\u969b\u9650\u306a\u3044\u5897\u52a0\uff08\u6307\u6570\u95a2\u6570\u7684\u5897\u52a0\uff09\u3092\u4e88\u6e2c\u3059\u308b\u3002\u3057\u304b\u3057\uff0c\u5b9f\u969b\u306b\u306f\u3044\u308d\u3044\u308d\u306a\u8981\u56e0\u306b\u3088\u308a\uff0c\u3053\u306e\u3088\u3046\u306a\u7121\u5236\u9650\u306a\u5897\u52a0\u306f\u7d9a\u304b\u306a\u3044\u3002<\/p>\n<p>\u4e0a\u306e\u30b0\u30e9\u30d5\u3067\u3082\uff0c1870\u5e74\u3042\u305f\u308a\u304b\u3089\uff0c\u30de\u30eb\u30b5\u30b9\u30fb\u30e2\u30c7\u30eb\u306e\u30b0\u30e9\u30d5\u306f\u5b9f\u969b\u306e\u30a2\u30e1\u30ea\u30ab\u306e\u4eba\u53e3\u30c7\u30fc\u30bf\u304b\u3089\u305a\u308c\u306f\u3058\u3081\uff0c1900\u5e74\u4ee5\u964d\u306f\u5168\u304f\u3042\u3063\u3066\u3044\u306a\u3044\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u30f4\u30a7\u30a2\u30d5\u30eb\u30b9\u30c8\u306b\u3088\u308b\u4fee\u6b63\u4eba\u53e3\u30e2\u30c7\u30eb\">\u30f4\u30a7\u30a2\u30d5\u30eb\u30b9\u30c8\u306b\u3088\u308b\u4fee\u6b63\u4eba\u53e3\u30e2\u30c7\u30eb<\/h3>\n<p>$$<br \/>\n\\frac{dN}{dt} = \\gamma N \\left(1 -\\frac{N}{N_{\\rm max}}\\right)<br \/>\n$$<\/p>\n<p>\u30f4\u30a7\u30a2\u30d5\u30eb\u30b9\u30c8\u306e\u4fee\u6b63\u4eba\u53e3\u30e2\u30c7\u30eb\u306e\u65b9\u7a0b\u5f0f\uff08\u30ed\u30b8\u30b9\u30c6\u30a3\u30c3\u30af\u65b9\u7a0b\u5f0f\uff09\u3092 <code>eqv<\/code> \u3068\u3057\uff0c\u3053\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092 <code>dsolve()<\/code> \u3067\u89e3\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[14]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">N<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Function<\/span><span class=\"p\">(<\/span><span class=\"s1\">'N'<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'gamma Nmax'<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">eqv<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">Derivative<\/span><span class=\"p\">(<\/span><span class=\"n\">N<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">),<\/span> <span class=\"n\">t<\/span><span class=\"p\">),<\/span> <span class=\"n\">gamma<\/span><span class=\"o\">*<\/span><span class=\"n\">N<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">N<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">Nmax<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">eqv<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[14]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{d}{d t} N{\\left(t \\right)} = \\gamma \\left(1 -\\frac{N{\\left(t \\right)}}{Nmax}\\right) N{\\left(t \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[15]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'t0 N0'<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">ansv<\/span> <span class=\"o\">=<\/span> <span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eqv<\/span><span class=\"p\">,<\/span> <span class=\"n\">N<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">),<\/span> <span class=\"n\">ics<\/span> <span class=\"o\">=<\/span> <span class=\"p\">{<\/span><span class=\"n\">N<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">subs<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">,<\/span><span class=\"n\">t0<\/span><span class=\"p\">):<\/span><span class=\"n\">N0<\/span><span class=\"p\">})<\/span>\r\n<span class=\"n\">ansv<\/span><span class=\"o\">.<\/span><span class=\"n\">simplify<\/span><span class=\"p\">()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[15]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle N{\\left(t \\right)} = \\frac{N_{0} Nmax e^{\\gamma t}}{N_{0} e^{\\gamma t} -\\left(N_{0} -Nmax\\right) e^{\\gamma t_{0}}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$\\displaystyle n(t) \\equiv \\frac{N(t)}{N_0}, \\ nmax \\equiv \\frac{N_{max}}{N_0}$ \u3068\u304a\u304f\u3068\uff0c\u4e0a\u306e\u89e3\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u300c\u898f\u683c\u5316\u3055\u308c\u305f\u300d\u89e3\u3068\u3057\u3066\u66f8\u3051\u307e\u3059\u3002<\/p>\n<p>$$n(t) = \\frac{nmax}{1 -(1 -nmax)\\, e^{\\gamma\\, (t_0 -t)}}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u53c2\u8003\uff1a\u7c73\u56fd\u306e\u4eba\u53e3\u3068\u30f4\u30a7\u30a2\u30d5\u30eb\u30b9\u30c8\u30e2\u30c7\u30eb\">\u53c2\u8003\uff1a\u7c73\u56fd\u306e\u4eba\u53e3\u3068\u30f4\u30a7\u30a2\u30d5\u30eb\u30b9\u30c8\u30e2\u30c7\u30eb<\/h4>\n<p>\u521d\u671f\u6761\u4ef6\u3092 $t_0 = 1790$\uff08\u5e74\uff09\u306e\u3068\u304d $N(t_0) = N_0 = 3.9$\uff08\u767e\u4e07\u4eba\uff09\u3068\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[16]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">t0<\/span> <span class=\"o\">=<\/span> <span class=\"n\">usa<\/span><span class=\"p\">[<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">]<\/span>\r\n<span class=\"n\">N0<\/span> <span class=\"o\">=<\/span> <span class=\"n\">usa<\/span><span class=\"p\">[<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">]<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u898f\u683c\u5316\u3055\u308c\u305f\u89e3 $n(t)$ \u3067\u306f $n(t_0) = 1$ \u3067\u3059\u3002$n(t)$ \u306e2\u3064\u306e\u30d1\u30e9\u30e1\u30fc\u30bf $nmax$ \u3068 $\\gamma$ \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\uff0c2\u3064\u306e\u5e74\u306e\u30c7\u30fc\u30bf\u304b\u3089\u6c42\u3081\u308b\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u307e\u305a\uff0c\u95a2\u6570 $n(t)$ \u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[17]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'nmax'<\/span><span class=\"p\">)<\/span>\r\n<span class=\"k\">def<\/span> <span class=\"nf\">n<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"n\">t0<\/span> <span class=\"o\">=<\/span> <span class=\"n\">usa<\/span><span class=\"p\">[<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">]<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"n\">nmax<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span> <span class=\"o\">-<\/span><span class=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">nmax<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"n\">exp<\/span><span class=\"p\">(<\/span><span class=\"n\">gamma<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"n\">t0<\/span><span class=\"o\">-<\/span><span class=\"n\">t<\/span><span class=\"p\">)))<\/span>\r\n<span class=\"n\">n<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[17]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{nmax}{-\\left(1 -nmax\\right) e^{\\gamma \\left(1790 -t\\right)} + 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u4f8b\u3068\u3057\u3066\uff0c<code>t1<\/code> = 1850\uff08\u5e74\uff09\u3068 <code>t2<\/code> = 1910\uff08\u5e74\uff09\u306e\u30c7\u30fc\u30bf\u3092\u4f7f\u3063\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[18]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">t1<\/span> <span class=\"o\">=<\/span> <span class=\"n\">usa<\/span><span class=\"p\">[<\/span><span class=\"mi\">6<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">]<\/span>\r\n<span class=\"n\">N1<\/span> <span class=\"o\">=<\/span> <span class=\"n\">usa<\/span><span class=\"p\">[<\/span><span class=\"mi\">6<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">]<\/span>\r\n<span class=\"n\">t1<\/span><span class=\"p\">,<\/span> <span class=\"n\">N1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[18]:<\/div>\n<div class=\"output_text output_subarea output_execute_result\">\n<pre>(1850, 23.2000000000000)<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[19]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">t2<\/span> <span class=\"o\">=<\/span> <span class=\"n\">usa<\/span><span class=\"p\">[<\/span><span class=\"mi\">12<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">]<\/span>\r\n<span class=\"n\">N2<\/span> <span class=\"o\">=<\/span> <span class=\"n\">usa<\/span><span class=\"p\">[<\/span><span class=\"mi\">12<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">]<\/span>\r\n<span class=\"n\">t2<\/span><span class=\"p\">,<\/span> <span class=\"n\">N2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[19]:<\/div>\n<div class=\"output_text output_subarea output_execute_result\">\n<pre>(1910, 92.0000000000000)<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u3053\u306e2\u7d44\u306e\u30c7\u30fc\u30bf\u3092\u4f7f\u3063\u3066\uff0c\u672a\u77e5\u5909\u6570 $nmax$ \u3068 $\\gamma$ \u306b\u3064\u3044\u3066\u306e\u9023\u7acb\u65b9\u7a0b\u5f0f\u3092\u3064\u304f\u308a\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[20]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">N1<\/span><span class=\"o\">\/<\/span><span class=\"n\">N0<\/span><span class=\"p\">,<\/span> <span class=\"n\">n<\/span><span class=\"p\">(<\/span><span class=\"n\">t1<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[20]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle 5.94871794871795 = \\frac{nmax}{-\\left(1 -nmax\\right) e^{-60 \\gamma} + 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[21]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">N2<\/span><span class=\"o\">\/<\/span><span class=\"n\">N0<\/span><span class=\"p\">,<\/span> <span class=\"n\">n<\/span><span class=\"p\">(<\/span><span class=\"n\">t2<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[21]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle 23.5897435897436 = \\frac{nmax}{-\\left(1 -nmax\\right) e^{-120 \\gamma} + 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u7c21\u5358\u306a\u4ee3\u6570\u65b9\u7a0b\u5f0f\u306b\u3059\u308b\u305f\u3081\uff0c$e^{-60 \\gamma} \\equiv T$ \u3068\u304a\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[22]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'T'<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">eqv1<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">N1<\/span><span class=\"o\">\/<\/span><span class=\"n\">N0<\/span><span class=\"p\">,<\/span> <span class=\"n\">n<\/span><span class=\"p\">(<\/span><span class=\"n\">t1<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">subs<\/span><span class=\"p\">(<\/span><span class=\"n\">exp<\/span><span class=\"p\">(<\/span><span class=\"o\">-<\/span><span class=\"mi\">60<\/span><span class=\"o\">*<\/span><span class=\"n\">gamma<\/span><span class=\"p\">),<\/span> <span class=\"n\">T<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">eqv1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[22]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle 5.94871794871795 = \\frac{nmax}{-T \\left(1 -nmax\\right) + 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[23]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">eqv2<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">N2<\/span><span class=\"o\">\/<\/span><span class=\"n\">N0<\/span><span class=\"p\">,<\/span> <span class=\"n\">n<\/span><span class=\"p\">(<\/span><span class=\"n\">t2<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">subs<\/span><span class=\"p\">(<\/span><span class=\"n\">exp<\/span><span class=\"p\">(<\/span><span class=\"o\">-<\/span><span class=\"mi\">60<\/span><span class=\"o\">*<\/span><span class=\"n\">gamma<\/span><span class=\"p\">),<\/span> <span class=\"n\">T<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">eqv2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[23]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle 23.5897435897436 = \\frac{nmax}{-T^{2} \\cdot \\left(1 -nmax\\right) + 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p><code>eqv1<\/code>, <code>eqv2<\/code> \u3092\u9023\u7acb\u3055\u305b\u3066 $nmax$, $T$ \u306b\u3064\u3044\u3066\u89e3\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[24]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">ans<\/span> <span class=\"o\">=<\/span> <span class=\"n\">solve<\/span><span class=\"p\">([<\/span><span class=\"n\">eqv1<\/span><span class=\"p\">,<\/span> <span class=\"n\">eqv2<\/span><span class=\"p\">],<\/span> <span class=\"p\">[<\/span><span class=\"n\">nmax<\/span><span class=\"p\">,<\/span> <span class=\"n\">T<\/span><span class=\"p\">])<\/span>\r\n<span class=\"n\">ans<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[24]:<\/div>\n<div class=\"output_text output_subarea output_execute_result\">\n<pre>[(49.9687003440904, 0.151115116017121)]<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u4e00\u822c\u306b\u306f\uff0c\u9023\u7acb\u65b9\u7a0b\u5f0f\u304b\u3089\u8907\u6570\u7d44\u306e\u89e3\u304c\u5f97\u3089\u308c\u308b\u53ef\u80fd\u6027\u304c\u3042\u308a\u307e\u3059\u304c\uff0c\u4eca\u56de\u306f1\u7d44\u306e\u307f\u3067\u3059\u3002<\/p>\n<p><code>ans[0]<\/code> \u3068\u3057\u30661\u7d44\u76ee\u306e\u89e3\u3092\u8868\u793a\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[25]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">ans<\/span><span class=\"p\">[<\/span><span class=\"mi\">0<\/span><span class=\"p\">]<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[25]:<\/div>\n<div class=\"output_text output_subarea output_execute_result\">\n<pre>(49.9687003440904, 0.151115116017121)<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u3053\u3046\u3057\u3066\u5f97\u3089\u308c\u305f $nmax$, $T$ \u306e\u5024\u3092\u5143\u306e $N_{max}$, $\\gamma$ \u306e\u5024\u306b\u306a\u304a\u3057\uff0c<br \/>\n\u30f4\u30a7\u30a2\u30d5\u30eb\u30b9\u30c8\u30e2\u30c7\u30eb\u306e\u89e3 $N(t)$ \u306b\u5165\u308c\u305f\u95a2\u6570\u3092 <code>Nv(t)<\/code> \u3068\u3057\u3066\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[26]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"nb\">print<\/span><span class=\"p\">(<\/span><span class=\"s1\">'Nmax = nmax*N0 ='<\/span><span class=\"p\">,<\/span> <span class=\"n\">ans<\/span><span class=\"p\">[<\/span><span class=\"mi\">0<\/span><span class=\"p\">][<\/span><span class=\"mi\">0<\/span><span class=\"p\">]<\/span><span class=\"o\">*<\/span><span class=\"n\">N0<\/span><span class=\"p\">)<\/span>\r\n<span class=\"nb\">print<\/span><span class=\"p\">(<\/span><span class=\"s1\">'gamma = -log(T)\/60 ='<\/span><span class=\"p\">,<\/span> <span class=\"o\">-<\/span><span class=\"n\">log<\/span><span class=\"p\">(<\/span><span class=\"n\">ans<\/span><span class=\"p\">[<\/span><span class=\"mi\">0<\/span><span class=\"p\">][<\/span><span class=\"mi\">1<\/span><span class=\"p\">])<\/span><span class=\"o\">\/<\/span><span class=\"mi\">60<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_subarea output_stream output_stdout output_text\">\n<pre>Nmax = nmax*N0 = 194.877931341953\r\ngamma = -log(T)\/60 = 0.0314952229147497\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[27]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">nmax0<\/span><span class=\"p\">,<\/span> <span class=\"n\">T0<\/span> <span class=\"o\">=<\/span> <span class=\"n\">ans<\/span><span class=\"p\">[<\/span><span class=\"mi\">0<\/span><span class=\"p\">]<\/span>\r\n\r\n<span class=\"k\">def<\/span> <span class=\"nf\">Nv<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"n\">t0<\/span> <span class=\"o\">=<\/span> <span class=\"n\">usa<\/span><span class=\"p\">[<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">]<\/span>\r\n    <span class=\"n\">N0<\/span> <span class=\"o\">=<\/span> <span class=\"n\">usa<\/span><span class=\"p\">[<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1<\/span><span class=\"p\">]<\/span>\r\n    <span class=\"n\">Nmax<\/span> <span class=\"o\">=<\/span> <span class=\"n\">nmax0<\/span> <span class=\"o\">*<\/span> <span class=\"n\">N0<\/span>\r\n    <span class=\"n\">gamma<\/span> <span class=\"o\">=<\/span> <span class=\"o\">-<\/span><span class=\"mi\">1<\/span><span class=\"o\">\/<\/span><span class=\"mi\">60<\/span> <span class=\"o\">*<\/span> <span class=\"n\">log<\/span><span class=\"p\">(<\/span><span class=\"n\">T0<\/span><span class=\"p\">)<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"n\">N0<\/span><span class=\"o\">*<\/span><span class=\"n\">Nmax<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"n\">N0<\/span><span class=\"o\">-<\/span><span class=\"p\">(<\/span><span class=\"n\">N0<\/span><span class=\"o\">-<\/span><span class=\"n\">Nmax<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"n\">exp<\/span><span class=\"p\">(<\/span><span class=\"n\">gamma<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"n\">t0<\/span><span class=\"o\">-<\/span><span class=\"n\">t<\/span><span class=\"p\">)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u30b0\u30e9\u30d5\u3092\u63cf\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[28]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">p4<\/span> <span class=\"o\">=<\/span> <span class=\"n\">plot<\/span><span class=\"p\">(<\/span><span class=\"n\">Nv<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">),<\/span> <span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1790<\/span><span class=\"p\">,<\/span> <span class=\"mi\">1940<\/span><span class=\"p\">),<\/span> <span class=\"s2\">\"\u30f4\u30a7\u30a2\u30d5\u30eb\u30b9\u30c8\u30e2\u30c7\u30eb\"<\/span><span class=\"p\">,<\/span> \r\n          <span class=\"n\">legend<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">True<\/span><span class=\"p\">,<\/span> \r\n          <span class=\"n\">xlabel<\/span> <span class=\"o\">=<\/span> <span class=\"s2\">\"\u5e74\"<\/span><span class=\"p\">,<\/span> <span class=\"n\">ylabel<\/span> <span class=\"o\">=<\/span> <span class=\"s2\">\"\u4eba\u53e3\uff08\u5358\u4f4d\uff1a\u767e\u4e07\u4eba\uff09\"<\/span><span class=\"p\">);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-9271\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/PMathC104.svg\" alt=\"\" width=\"640\" height=\"427\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>3\u3064\u306e\u30b0\u30e9\u30d5\u3092\u4e00\u7dd2\u306b\u8868\u793a\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[29]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">p5<\/span> <span class=\"o\">=<\/span> <span class=\"n\">p3<\/span><span class=\"o\">+<\/span><span class=\"n\">p4<\/span>\r\n<span class=\"n\">p5<\/span><span class=\"o\">.<\/span><span class=\"n\">show<\/span><span class=\"p\">();<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt\"><\/div>\n<div class=\"output_svg output_subarea \"><\/div>\n<\/div>\n<div class=\"output_area\">\n<div class=\"output_svg output_subarea \"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-9272\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/PMathC105.svg\" alt=\"\" width=\"640\" height=\"427\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u3053\u306e\u30b0\u30e9\u30d5\u3092\u307f\u308b\u3068\uff0c\uff08$N_{max}$ \u3068 $\\gamma$ \u3092\u3046\u307e\u304f\u3042\u308f\u305b\u308b\u3053\u3068\u306b\u3088\u3063\u3066\uff09<strong>\u30f4\u30a7\u30a2\u30d5\u30eb\u30b9\u30c8\u30e2\u30c7\u30eb<\/strong>\u306e\u307b\u3046\u304c\uff0c\u3053\u306e\u5e74\u4ee3\u306e\u7c73\u56fd\u306e\u4eba\u53e3\u3092\u3088\u304f\u3042\u3089\u308f\u3057\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"1\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\u3068\u7a4d\u5206\u56e0\u5b50\u6cd5\">1\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\u3068\u7a4d\u5206\u56e0\u5b50\u6cd5<\/h3>\n<p>SymPy \u3067\u306f\uff0c1\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f\uff0c\u7279\u306b\u7a4d\u5206\u56e0\u5b50\u6cd5\u3092 coding \u3057\u306a\u304f\u3066\u3082\uff0c<code>dsolve()<\/code> \u95a2\u6570\u3067\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<h4 id=\"\u4f8b\u984c\">\u4f8b\u984c<\/h4>\n<p>$$ \\frac{dy}{dx} + \\frac{y}{x} = \\frac{\\sin x}{x} $$<\/p>\n<p>SymPy \u3067\u306e\u4e09\u89d2\u95a2\u6570 $\\sin x, \\cos x, \\tan x$ \u306f <code>sin(x)<\/code>, <code>cos(x)<\/code>, <code>tan(x)<\/code> \u3068\u66f8\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[30]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">y<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Function<\/span><span class=\"p\">(<\/span><span class=\"s1\">'y'<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">eq<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">Derivative<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">+<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"n\">sin<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">eq<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[30]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{d}{d x} y{\\left(x \\right)} + \\frac{y{\\left(x \\right)}}{x} = \\frac{\\sin{\\left(x \\right)}}{x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[31]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eq<\/span><span class=\"p\">,<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[31]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = \\frac{C_{1} -\\cos{\\left(x \\right)}}{x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u53c2\u8003\uff1a\u3042\u3048\u3066\u7a4d\u5206\u56e0\u5b50\u6cd5\u306e\u516c\u5f0f\u3067...\">\u53c2\u8003\uff1a\u3042\u3048\u3066\u7a4d\u5206\u56e0\u5b50\u6cd5\u306e\u516c\u5f0f\u3067&#8230;<\/h4>\n<p>\u3053\u306e\u4f8b\u984c\u3092\u3042\u3048\u3066\u7a4d\u5206\u56e0\u5b50\u6cd5\u306e\u516c\u5f0f\u306b\u5f93\u3063\u3066\u3084\u3063\u3066\u307f\u308b\u3002<\/p>\n<p>\u4e00\u822c\u7684\u306a1\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\uff08\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u683c\u597d\u3057\u3066\u3044\u308b\u3082\u306e\uff09\u3092<\/p>\n<p>$$\\frac{dy}{dx} + P(x) y = Q(x)$$<\/p>\n<p>\u3068\u66f8\u304f\u3068\uff0c\u7a4d\u5206\u56e0\u5b50 $g(x)$ \u306f<\/p>\n<p>$$g(x) = \\exp \\left\\{\\int^x P(x&#8217;)\\,dx&#8217; \\right\\}$$<\/p>\n<p>\u89e3\u306f<\/p>\n<p>$$y = \\frac{1}{g(x)} \\left\\{\\int^x g(x&#8217;) Q(x&#8217;) \\, dx&#8217; + C \\right\\}$$<\/p>\n<p>\u3053\u306e\u4f8b\u984c\u3067\u306f<\/p>\n<p>$$P(x) = \\frac{1}{x}, \\ Q(x) = \\frac{\\sin x}{x}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[32]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"k\">def<\/span> <span class=\"nf\">P<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"mi\">1<\/span><span class=\"o\">\/<\/span><span class=\"n\">x<\/span>\r\n\r\n<span class=\"k\">def<\/span> <span class=\"nf\">Q<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"n\">sin<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"n\">x<\/span>\r\n\r\n<span class=\"n\">g<\/span> <span class=\"o\">=<\/span> <span class=\"n\">exp<\/span><span class=\"p\">(<\/span><span class=\"n\">integrate<\/span><span class=\"p\">(<\/span><span class=\"n\">P<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">g<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[32]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle x$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[33]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">y<\/span> <span class=\"o\">=<\/span> <span class=\"mi\">1<\/span><span class=\"o\">\/<\/span><span class=\"n\">g<\/span> <span class=\"o\">*<\/span> <span class=\"p\">(<\/span><span class=\"n\">integrate<\/span><span class=\"p\">(<\/span><span class=\"n\">g<\/span> <span class=\"o\">*<\/span> <span class=\"n\">Q<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">)<\/span> <span class=\"o\">+<\/span> <span class=\"n\">C<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">y<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[33]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{C -\\cos{\\left(x \\right)}}{x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u6700\u3082\u7c21\u5358\u306a\u5b9a\u6570\u4fc2\u65702\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\">\u6700\u3082\u7c21\u5358\u306a\u5b9a\u6570\u4fc2\u65702\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h3>\n<p>\u6700\u3082\u7c21\u5358\u306a\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f $y&#8221; + y = 0$ \u3068 $y&#8221; -y = 0$ \u3092\u89e3\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[34]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">y<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Function<\/span><span class=\"p\">(<\/span><span class=\"s1\">'y'<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">eq1<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Derivative<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"mi\">2<\/span><span class=\"p\">)<\/span> <span class=\"o\">+<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">eq1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[34]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} + \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[35]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eq1<\/span><span class=\"p\">,<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[35]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = C_{1} \\sin{\\left(x \\right)} + C_{2} \\cos{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[36]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">eq2<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Derivative<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"mi\">2<\/span><span class=\"p\">)<\/span> <span class=\"o\">-<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">eq2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[36]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle -y{\\left(x \\right)} + \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[37]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eq2<\/span><span class=\"p\">,<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[37]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = C_{1} e^{-x} + C_{2} e^{x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u53c2\u8003\uff1a\u810a\u9ac4\u53cd\u5c04\u306b\u3088\u3089\u305a\u306b\u89e3\u304f\">\u53c2\u8003\uff1a\u810a\u9ac4\u53cd\u5c04\u306b\u3088\u3089\u305a\u306b\u89e3\u304f<\/h4>\n<p>$y&#8221; + y = 0$ \u3092\u810a\u9ac4\u53cd\u5c04\u306b\u3088\u3089\u305a\u306b\u89e3\u304f\u3002\u4e21\u8fba\u306b $2 y&#8217;$ \u3092\u304b\u3051\u3066\u6574\u7406\u3059\u308b\u3068<\/p>\n<p>$$\\left(\\left(y&#8217;\\right)^2 + y^2\\right)&#8217; = 0$$<\/p>\n<p>\u5fae\u5206\u3057\u3066\u30bc\u30ed\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u304b\u3063\u3053\u306e\u4e2d\u8eab\u306f\u5b9a\u6570\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u306a\u306e\u3067\uff0c<\/p>\n<p>$$ \\left(y&#8217;\\right)^2 + y^2 = \\mbox{const.} \\equiv a^2$$<\/p>\n<p>\u3068\u304a\u3051\u308b\u3002\u3064\u307e\u308a<\/p>\n<p>$$\\frac{dy}{dx} = \\pm \\sqrt{a^2 -y^2}$$<\/p>\n<p>\u3092\u89e3\u3051\u3070\u3088\u3044\u3002\u307e\u305a\u306f $+$ \u306e\u5f0f\u304b\u3089&#8230;<\/p>\n<p>$\\displaystyle \\frac{dy}{dx} = + \\sqrt{a^2 &#8211; y^2}$ \u3092\u89e3\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u6ce8\u610f\u3059\u3079\u304d\u3053\u3068\u306f\uff0c$\\displaystyle \\sqrt{a^2}$ \u3067\u3059\u3002SymPy \u304c\u3069\u306e\u3088\u3046\u306a\u7b54\u3048\u3092\u8fd4\u3059\u304b\u898b\u3066\u307f\u308b\u3068&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[38]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'a'<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[38]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\sqrt{a^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$\\displaystyle \\sqrt{a^2} = |a|$ \u3067\u3059\u304c SymPy \u306f\u305d\u306e\u307e\u307e\u8fd4\u3057\u307e\u3059\u3002\u3067\u306f $a &gt; 0$ \u3068\u5ba3\u8a00\u3059\u308b\u3068&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[39]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'a'<\/span><span class=\"p\">,<\/span> <span class=\"n\">positive<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">True<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[39]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle a$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u53c2\u8003\u307e\u3067\u306b\uff0c$a &lt; 0$ \u3068\u5ba3\u8a00\u3059\u308c\u3070&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[40]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'a'<\/span><span class=\"p\">,<\/span> <span class=\"n\">negative<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">True<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[40]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle &#8211; a$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u4ee5\u4e0b\u3067\u306f\u7c21\u5358\u306e\u305f\u3081\u306b\uff0c$a &gt; 0$ \u3068\u3057\u3066\u8a71\u3092\u9032\u3081\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[41]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u4e00\u822c\u6027\u3092\u5931\u3046\u3053\u3068\u306a\u304f\u7b2c1\u306e\u7a4d\u5206\u5b9a\u6570\u306e\u5e73\u65b9\u6839 a \u3092\u6b63\u306b\u3067\u304d\u308b<\/span>\r\n<span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'a'<\/span><span class=\"p\">,<\/span> <span class=\"n\">positive<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">True<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">eq1<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">Derivative<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span> <span class=\"o\">-<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">eq1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[41]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{d}{d x} y{\\left(x \\right)} = \\sqrt{a^{2} -y^{2}{\\left(x \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[42]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eq1<\/span><span class=\"p\">,<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[42]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = a \\sin{\\left(C_{1} + x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u6b21\u306b $\\displaystyle \\frac{dy}{dx} = -\\sqrt{a^2 -y^2}$ \u3092\u89e3\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[43]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">eq2<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">Derivative<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"o\">-<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span> <span class=\"o\">-<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">eq2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[43]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{d}{d x} y{\\left(x \\right)} = -\\sqrt{a^{2} -y^{2}{\\left(x \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[44]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eq2<\/span><span class=\"p\">,<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[44]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = a \\sin{\\left(C_{1} -x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$y(x) = a \\sin\\left(C_1 -x\\right) = -a \\sin(x -C_1)$<br \/>\n\u3068\u3044\u3046\u3053\u3068\u3067 2 \u3064\u306e\u89e3\u3092\u307e\u3068\u3081\u308b\u3068\uff0c\u7a4d\u5206\u5b9a\u6570\u3042\u3089\u305f\u3081\u3066 $\\pm a \\Rightarrow C_2, \\pm C_1 \\Rightarrow C_1$ \u3068\u304a\u3044\u3066<\/p>\n<p>\\begin{eqnarray}<br \/>\ny &amp;=&amp; C_2 \\sin(x + C_1) \\\\<br \/>\n&amp;=&amp; C_2 \\left(\\sin x \\cos C_1 + \\cos x \\sin C_1 \\right) \\\\<br \/>\n&amp;=&amp; (C_2 \\cos C_1) \\sin x + (C_2 \\sin C_1) \\cos x \\\\<br \/>\n&amp;\\equiv&amp; A \\cos x + B \\sin x\\\\<br \/>\n\\mbox{\u3053\u3053\u3067} \\ \\ A &amp;\\equiv&amp; C_2 \\sin C_1 \\\\<br \/>\nB &amp;\\equiv&amp; C_2 \\cos C_1<br \/>\n\\end{eqnarray}<\/p>\n<p>\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3068\u3044\u3046\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u6700\u3082\u7c21\u5358\u306a\u5b9a\u6570\u4fc2\u65702\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\uff1a\u7d9a\u304d\">\u6700\u3082\u7c21\u5358\u306a\u5b9a\u6570\u4fc2\u65702\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\uff1a\u7d9a\u304d<\/h3>\n<p>$y&#8221; + K y = 0$ \u3042\u308b\u3044\u306f\u79fb\u9805\u3057\u3066 $y&#8221; = -K y$ \u3092\u89e3\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"$K-&gt;-0$-\u306e\u5834\u5408\">$K &gt; 0$ \u306e\u5834\u5408<\/h4>\n<p>$K &gt; 0$ \u306f SymPy \u3067\u306f <code>var('K', positive = True)<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[45]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">y<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Function<\/span><span class=\"p\">(<\/span><span class=\"s1\">'y'<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'x'<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'K'<\/span><span class=\"p\">,<\/span> <span class=\"n\">positive<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">True<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">eq1<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">Derivative<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"mi\">2<\/span><span class=\"p\">),<\/span> <span class=\"o\">-<\/span><span class=\"n\">K<\/span><span class=\"o\">*<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">eq1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[45]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)} = -K y{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[46]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eq1<\/span><span class=\"p\">,<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[46]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = C_{1} \\sin{\\left(\\sqrt{K} x \\right)} + C_{2} \\cos{\\left(\\sqrt{K} x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"$K-&lt;-0$-\u306e\u5834\u5408\">$K &lt; 0$ \u306e\u5834\u5408<\/h4>\n<p>$K &lt; 0$ \u306f SymPy \u3067\u306f <code>var('K', negative = True)<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[47]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'K'<\/span><span class=\"p\">,<\/span> <span class=\"n\">negative<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">True<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">eq2<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">Derivative<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"mi\">2<\/span><span class=\"p\">),<\/span> <span class=\"o\">-<\/span><span class=\"n\">K<\/span><span class=\"o\">*<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">eq2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[47]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)} = -K y{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[48]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eq2<\/span><span class=\"p\">,<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[48]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = C_{1} e^{-x \\sqrt{-K}} + C_{2} e^{x \\sqrt{-K}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"$K-=-0$-\u306e\u5834\u5408\">$K = 0$ \u306e\u5834\u5408<\/h4>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[49]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">eq3<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">Derivative<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"mi\">2<\/span><span class=\"p\">),<\/span> <span class=\"mi\">0<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">eq3<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[49]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)} = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[50]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eq3<\/span><span class=\"p\">,<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[50]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = C_{1} + C_{2} x$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u4eba\u985e\u306e\u81f3\u5b9d\uff1a\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\">\u4eba\u985e\u306e\u81f3\u5b9d\uff1a\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f<\/h3>\n<p>SymPy \u304c\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u3092\u77e5\u3063\u3066\u3044\u308b\u304b\u78ba\u8a8d\u3059\u308b\u3002<\/p>\n<p>SymPy \u3067\u306f<\/p>\n<ul>\n<li>\u865a\u6570\u5358\u4f4d $i$ \u306f <code>I<\/code><\/li>\n<li>\u5186\u5468\u7387 $\\pi$ \u306f <code>pi<\/code><\/li>\n<li>\u30cd\u30a4\u30d4\u30a2\u6570 $e$ \u306f <code>E<\/code><\/li>\n<\/ul>\n<p>\u3067\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u6307\u6570\u95a2\u6570 $e^{i x}$ \u3092 <code>.rewrite(cos)<\/code> \u3092\u3064\u3051\u3066\u4e09\u89d2\u95a2\u6570\u3067\u66f8\u304d\u76f4\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[51]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">exp<\/span><span class=\"p\">(<\/span><span class=\"n\">I<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> \r\n   <span class=\"n\">exp<\/span><span class=\"p\">(<\/span><span class=\"n\">I<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">rewrite<\/span><span class=\"p\">(<\/span><span class=\"n\">cos<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[51]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle e^{i x} = i \\sin{\\left(x \\right)} + \\cos{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u30aa\u30a4\u30e9\u30fc\u306e\u9805\u5f0f $e^{i x} = \\cos x + i \\sin x$ \u304c\u78ba\u8a8d\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u30aa\u30a4\u30e9\u30fc\u306e\u7b49\u5f0f\">\u30aa\u30a4\u30e9\u30fc\u306e\u7b49\u5f0f<\/h4>\n<p>$x = \\pi$ \u306e\u3068\u304d\uff0c$e^{i \\pi} + 1 = 0$\u3002\u6307\u6570\u95a2\u6570 <code>exp()<\/code> \u3067\u306f\u306a\u304f\uff0c\u30cd\u30a4\u30d4\u30a2\u6570 <code>E<\/code> \u3092\u4f7f\u3063\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[52]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">E<\/span><span class=\"o\">**<\/span><span class=\"p\">(<\/span><span class=\"n\">I<\/span><span class=\"o\">*<\/span><span class=\"n\">pi<\/span><span class=\"p\">)<\/span> <span class=\"o\">+<\/span> <span class=\"mi\">1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[52]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u5de6\u8fba\u3068\u53f3\u8fba\u304c\u7b49\u3057\u3044\u304b\u3069\u3046\u304b\u3092\u78ba\u8a8d\u3059\u308b\u3068\u304d\u306b\u306f <code>==<\/code> \u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[53]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">E<\/span><span class=\"o\">**<\/span><span class=\"p\">(<\/span><span class=\"n\">I<\/span><span class=\"o\">*<\/span><span class=\"n\">pi<\/span><span class=\"p\">)<\/span> <span class=\"o\">+<\/span> <span class=\"mi\">1<\/span> <span class=\"o\">==<\/span> <span class=\"mi\">0<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[53]:<\/div>\n<div class=\"output_text output_subarea output_execute_result\">\n<pre>True<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u304b\u3089\u307f\u305f\u4e09\u89d2\u95a2\u6570\u3068\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u95a2\u4fc2\">\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u304b\u3089\u307f\u305f\u4e09\u89d2\u95a2\u6570\u3068\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u95a2\u4fc2<\/h4>\n<p>\u4e09\u89d2\u95a2\u6570 $\\cos x, \\sin x$ \u3092 <code>.rewrite(exp)<\/code> \u3067\uff08\u8907\u7d20\uff09\u6307\u6570\u95a2\u6570\u8868\u793a\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[54]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">cos<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">rewrite<\/span><span class=\"p\">(<\/span><span class=\"n\">exp<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[54]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{e^{i x}}{2} + \\frac{e^{-i x}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[55]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">sin<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">rewrite<\/span><span class=\"p\">(<\/span><span class=\"n\">exp<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[55]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle -\\frac{i \\left(e^{i x} -e^{-i x}\\right)}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u53cc\u66f2\u7dda\u95a2\u6570 $\\cosh x, \\sinh x$ \u3082\u6307\u6570\u95a2\u6570\u8868\u793a\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[56]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">cosh<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">rewrite<\/span><span class=\"p\">(<\/span><span class=\"n\">exp<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[56]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{e^{x}}{2} + \\frac{e^{-x}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[57]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">sinh<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">rewrite<\/span><span class=\"p\">(<\/span><span class=\"n\">exp<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[57]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{e^{x}}{2} -\\frac{e^{-x}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>&#8230;\u3068\u3044\u3046\u308f\u3051\u3067\uff0c\u4ee5\u4e0b\u306e\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\cos x &amp;=&amp; \\frac{e^{ix} + e^{-ix}}{2} \\\\<br \/>\n\\sin x &amp;=&amp; \\frac{e^{ix} -e^{-ix}}{2 i} \\\\<br \/>\n\\cosh x &amp;=&amp; \\frac{e^x + e^{-x}}{2} \\\\<br \/>\n\\sinh x &amp;=&amp; \\frac{e^x -e^{-x}}{2}<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3055\u3066\uff0c\u4e09\u89d2\u95a2\u6570\u3084\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5909\u6570\uff08\u5f15\u6570\uff09\u304c\u865a\u6570\u3067\u3082\u3044\u3044\u306e\u3060\u3068\u62e1\u5f35\u3059\u308b\u3068&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[58]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">cosh<\/span><span class=\"p\">(<\/span><span class=\"n\">I<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[58]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\cos{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[59]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">sinh<\/span><span class=\"p\">(<\/span><span class=\"n\">I<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[59]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle i \\sin{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[60]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">cos<\/span><span class=\"p\">(<\/span><span class=\"n\">I<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[60]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\cosh{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[61]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">sin<\/span><span class=\"p\">(<\/span><span class=\"n\">I<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[61]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle i \\sinh{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>&#8230; \u3068\u3044\u3046\u3053\u3068\u3067\u4ee5\u4e0b\u306e\u3053\u3068\u3092\u78ba\u8a8d\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\cosh (i x) &amp;=&amp; \\cos x \\\\<br \/>\n\\sinh (i x) &amp;=&amp; i \\sin x \\\\<br \/>\n\\cos (i x) &amp;=&amp; \\cosh x \\\\<br \/>\n\\sin (i x) &amp;=&amp; i \\sinh x<br \/>\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u4f8b\u984c\">\u4f8b\u984c<\/h4>\n<p>\u5fae\u5206\u65b9\u7a0b\u5f0f $y&#8221; + K y = 0$ \u306e $K &gt; 0$ \u306e\u5834\u5408\u306e\u89e3\u306f<\/p>\n<p>\\begin{eqnarray}<br \/>\ny &amp;=&amp; A \\cos\\left( \\sqrt{K} x\\right) + \\frac{B}{\\sqrt{K}} \\sin\\left( \\sqrt{K} x\\right)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002\u3053\u306e\u3053\u3068\u3092\u4f7f\u3063\u3066\uff0c$K&lt;0$ \u306e\u5834\u5408\u3068 $K = 0$ \u306e\u5834\u5408\u306e\u89e3\u3092\u4e0a\u8a18\u306e\u89e3\u304b\u3089\u76f4\u63a5\u6c42\u3081\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[62]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u5ff5\u306e\u305f\u3081\u306b\u5b9a\u6570\u306e\u5ba3\u8a00<\/span>\r\n<span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'A B x'<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'K'<\/span><span class=\"p\">,<\/span> <span class=\"n\">positve<\/span> <span class=\"o\">=<\/span><span class=\"kc\">True<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">y1<\/span> <span class=\"o\">=<\/span> <span class=\"n\">A<\/span><span class=\"o\">*<\/span><span class=\"n\">cos<\/span><span class=\"p\">(<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">K<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span> <span class=\"o\">+<\/span> <span class=\"n\">B<\/span><span class=\"o\">\/<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">K<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"n\">sin<\/span><span class=\"p\">(<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">K<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">y1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[62]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle A \\cos{\\left(\\sqrt{K} x \\right)} + \\frac{B \\sin{\\left(\\sqrt{K} x \\right)}}{\\sqrt{K}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h5 id=\"$K-&lt;-0$-\u306e\u5834\u5408\">$K &lt; 0$ \u306e\u5834\u5408<\/h5>\n<p><code>.subs(K, -abs(K))<\/code> \u3067 $K$ \u306b $-|K|$ \u3092\u4ee3\u5165\u3059\u308b\u3068&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[63]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">y2<\/span> <span class=\"o\">=<\/span> <span class=\"n\">y1<\/span><span class=\"o\">.<\/span><span class=\"n\">subs<\/span><span class=\"p\">(<\/span><span class=\"n\">K<\/span><span class=\"p\">,<\/span> <span class=\"o\">-<\/span><span class=\"nb\">abs<\/span><span class=\"p\">(<\/span><span class=\"n\">K<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">y2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[63]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle A \\cosh{\\left(x \\sqrt{\\left|{K}\\right|} \\right)} + \\frac{B \\sinh{\\left(x \\sqrt{\\left|{K}\\right|} \\right)}}{\\sqrt{\\left|{K}\\right|}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h5 id=\"$K-=-0$-\u306e\u5834\u5408\">$K = 0$ \u306e\u5834\u5408<\/h5>\n<p>$K \\rightarrow 0$ \u306e\u6975\u9650\u3092\u3068\u308a\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[64]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">Limit<\/span><span class=\"p\">(<\/span><span class=\"n\">y1<\/span><span class=\"p\">,<\/span> <span class=\"n\">K<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">),<\/span> \r\n   <span class=\"n\">Limit<\/span><span class=\"p\">(<\/span><span class=\"n\">y1<\/span><span class=\"p\">,<\/span> <span class=\"n\">K<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">doit<\/span><span class=\"p\">())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[64]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\lim_{K \\to 0^+}\\left(A \\cos{\\left(\\sqrt{K} x \\right)} + \\frac{B \\sin{\\left(\\sqrt{K} x \\right)}}{\\sqrt{K}}\\right) = A + B x$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>&#8230; \u3068\u3044\u3046\u3053\u3068\u3067\uff0c$K&gt;0$ \u306e\u5834\u5408\u306e\u89e3\u30921\u3064\u304a\u307c\u3048\u308c\u3070\uff0c$K &lt; 0$ \u306e\u5834\u5408\u3082\uff0c$ K =0$ \u306e\u5834\u5408\u3082\u305d\u308c\u304b\u3089\u308f\u304b\u308b\uff0c\u3068\u3044\u3046\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u540c\u6b21\u65b9\u7a0b\u5f0f\">\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u540c\u6b21\u65b9\u7a0b\u5f0f<\/h3>\n<p>\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\uff08\u540c\u6b21\u65b9\u7a0b\u5f0f\uff09\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<\/p>\n<p>$$ \\frac{d^2 y}{dx^2} + 2 b \\frac{dy}{dx} + cy = 0$$<\/p>\n<p>$b, c$ \u306f\u5b9a\u6570\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[65]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">y<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Function<\/span><span class=\"p\">(<\/span><span class=\"s1\">'y'<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'x'<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'b c'<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">eq<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Derivative<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"mi\">2<\/span><span class=\"p\">)<\/span> <span class=\"o\">+<\/span> <span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">b<\/span><span class=\"o\">*<\/span><span class=\"n\">Derivative<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">)<\/span> <span class=\"o\">+<\/span> <span class=\"n\">c<\/span><span class=\"o\">*<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">eq<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[65]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle 2 b \\frac{d}{d x} y{\\left(x \\right)} + c y{\\left(x \\right)} + \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u7279\u306b\u4eee\u5b9a\u306a\u3057\u3067 <code>eq = 0<\/code> \u3092 <code>dsolve()<\/code> \u3057\u305f\u3068\u304d\u306e\u89e3\u3002\u30eb\u30fc\u30c8\u306e\u4e2d\u8eab $b^2 -c &gt;0$ \u3092\u4eee\u5b9a\u3057\u305f\u3068\u304d\u306e\u89e3\u306b\u306a\u3063\u3066\u3044\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[66]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">ans1<\/span> <span class=\"o\">=<\/span> <span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eq<\/span><span class=\"p\">,<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">ans1<\/span><span class=\"o\">.<\/span><span class=\"n\">simplify<\/span><span class=\"p\">()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[66]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = \\left(C_{1} e^{x \\sqrt{b^{2} -c}} + C_{2} e^{-x \\sqrt{b^{2} -c}}\\right) e^{-b x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u3072\u3068\u3064\u3060\u3051\u306e\u89e3\u3067\u3059\u307e\u3059\u306b\u306f\u2026\">\u3072\u3068\u3064\u3060\u3051\u306e\u89e3\u3067\u3059\u307e\u3059\u306b\u306f\u2026<\/h4>\n<p>\u4ee5\u4e0b\u306e\u8868\u73fe\u3072\u3068\u3064\u30673\u3064\u306e\u5834\u5408\u306e\u5168\u3066\u306b\u5bfe\u5fdc\u3057\u305f\u89e3\u306b\u306a\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[67]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"k\">def<\/span> <span class=\"nf\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">mu<\/span><span class=\"p\">,<\/span> <span class=\"n\">x<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/span> <span class=\"n\">exp<\/span><span class=\"p\">(<\/span><span class=\"o\">-<\/span><span class=\"n\">b<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"p\">(<\/span><span class=\"n\">A<\/span><span class=\"o\">*<\/span><span class=\"n\">cos<\/span><span class=\"p\">(<\/span><span class=\"n\">mu<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span> <span class=\"o\">+<\/span> <span class=\"n\">B<\/span><span class=\"o\">\/<\/span><span class=\"n\">mu<\/span> <span class=\"o\">*<\/span> <span class=\"n\">sin<\/span><span class=\"p\">(<\/span><span class=\"n\">mu<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">mu<\/span><span class=\"p\">,<\/span> <span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[67]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\left(A \\cos{\\left(\\mu x \\right)} + \\frac{B \\sin{\\left(\\mu x \\right)}}{\\mu}\\right) e^{-b x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$c-b^2 &gt; 0$ \u306e\u5834\u5408\u306f $\\mu \\equiv \\sqrt{c-b^2}$ \u3068\u3057\u3066&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[68]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">c<\/span> <span class=\"o\">-<\/span> <span class=\"n\">b<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[68]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\left(A \\cos{\\left(x \\sqrt{-b^{2} + c} \\right)} + \\frac{B \\sin{\\left(x \\sqrt{-b^{2} + c} \\right)}}{\\sqrt{-b^{2} + c}}\\right) e^{-b x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$c-b^2 &lt; 0$ \u306e\u5834\u5408\u306f $\\mu \\equiv \\sqrt{c-b^2} = i \\sqrt{b^2-c}$ \u3068\u3057\u3066&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[69]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">I<\/span><span class=\"o\">*<\/span><span class=\"n\">sqrt<\/span><span class=\"p\">(<\/span><span class=\"n\">b<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span><span class=\"o\">-<\/span><span class=\"n\">c<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[69]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\left(A \\cosh{\\left(x \\sqrt{b^{2} -c} \\right)} + \\frac{B \\sinh{\\left(x \\sqrt{b^{2} -c} \\right)}}{\\sqrt{b^{2} -c}}\\right) e^{-b x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u6700\u5f8c\u306b $c-b^2 = 0$ \u306e\u5834\u5408\u306f $\\mu \\rightarrow 0$ \u306e\u6975\u9650\u3092\u3068\u3063\u3066&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[70]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">limit<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">mu<\/span><span class=\"p\">,<\/span> <span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">mu<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[70]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\left(A + B x\\right) e^{-b x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h3 id=\"\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\">\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f<\/h3>\n<p>\u4eba\u529b\u3067\u89e3\u304f\u969b\u306b\u306f\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f\u3068\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u3068\u3067\u306f\uff0c\u89e3\u304f\u624b\u9593\u304c\u305a\u3044\u3076\u3093\u9055\u3063\u305f\u304c\uff0cSymPy \u3067\u306f\u3069\u3061\u3089\u3082\u540c\u3058\u3002<code>dsolve()<\/code> \u3092\u4f7f\u3046\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u4f8b\u984c\">\u4f8b\u984c<\/h4>\n<p>\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f $y&#8221; + a^2 y = \\sin b x$ \u3092\u89e3\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[71]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">y<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Function<\/span><span class=\"p\">(<\/span><span class=\"s1\">'y'<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'x'<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'a b'<\/span><span class=\"p\">,<\/span> <span class=\"n\">positive<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">True<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">eq<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">Derivative<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"mi\">2<\/span><span class=\"p\">)<\/span> <span class=\"o\">+<\/span> <span class=\"n\">a<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span> <span class=\"o\">*<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">sin<\/span><span class=\"p\">(<\/span><span class=\"n\">b<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">eq<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[71]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle a^{2} y{\\left(x \\right)} + \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)} = \\sin{\\left(b x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[72]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eq<\/span><span class=\"p\">,<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[72]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = C_{1} \\sin{\\left(a x \\right)} + C_{2} \\cos{\\left(a x \\right)} + \\frac{\\sin{\\left(b x \\right)}}{a^{2} -b^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$a = b$ \u306e\u5834\u5408\u306f\u4e0a\u8a18\u306e\u89e3\u306e\u5206\u6bcd\u304c\u30bc\u30ed\u306b\u306a\u3063\u3066\u3057\u307e\u3046\u306e\u3067\uff0c\u5225\u9014\u8a08\u7b97\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002<\/p>\n<p>\u3053\u306e\u72b6\u6cc1\u306f\uff0c\u529b\u5b66\u3067\u306f\u56fa\u6709\u632f\u52d5\u6570 $a$<br \/>\n\u306b\u7b49\u3057\u3044\u632f\u52d5\u6570\u306e\u5916\u529b\u304c\u52a0\u3048\u3089\u308c\u305f\u6642\u306b\u8d77\u3053\u308b\u300c\u5171\u9cf4\uff08\u5171\u632f\uff09\u300d\u3068\u547c\u3070\u308c\u308b\u73fe\u8c61\u3067\u3042\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[73]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">b<\/span> <span class=\"o\">=<\/span> <span class=\"n\">a<\/span>\r\n<span class=\"n\">eq2<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">Derivative<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">x<\/span><span class=\"p\">,<\/span> <span class=\"mi\">2<\/span><span class=\"p\">)<\/span> <span class=\"o\">+<\/span> <span class=\"n\">a<\/span><span class=\"o\">**<\/span><span class=\"mi\">2<\/span> <span class=\"o\">*<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">),<\/span> <span class=\"n\">sin<\/span><span class=\"p\">(<\/span><span class=\"n\">b<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">eq2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[73]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle a^{2} y{\\left(x \\right)} + \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)} = \\sin{\\left(a x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[74]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eq2<\/span><span class=\"p\">,<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[74]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = C_{2} \\sin{\\left(a x \\right)} + \\left(C_{1} -\\frac{x}{2 a}\\right) \\cos{\\left(a x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u7a4d\u5206\u5b9a\u6570 $C_1, C_2 $\u304c\u3064\u304f\u306e\u306f\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\u3002<\/p>\n<p>\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3<br \/>\n$\\displaystyle -\\frac{x}{2a} \\cos (a x)$ \u306f\uff0c\u632f\u5e45\u304c $x$ \u306b\u6bd4\u4f8b\u3057\u3066\u5358\u8abf\u5897\u52a0\u3057\u3066\u3044\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<h4 id=\"\u3042\u3048\u3066\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u30ed\u30f3\u30b9\u30ad\u30a2\u30f3\u3092\u4f7f\u3063\u3066\u7279\u6b8a\u89e3\u3092\u6c42\u3081\u308b\">\u3042\u3048\u3066\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u30ed\u30f3\u30b9\u30ad\u30a2\u30f3\u3092\u4f7f\u3063\u305f\u516c\u5f0f\u3067\u7279\u6b8a\u89e3\u3092\u6c42\u3081\u308b<\/h4>\n<p>\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u3082 <code>dsolve()<\/code> \u3067\u89e3\u304c\u3082\u3068\u307e\u3063\u305f\u308f\u3051\u3060\u304c\uff0c\u305d\u3053\u3092\u3042\u3048\u3066\u30ed\u30f3\u30b9\u30ad\u30a2\u30f3\u3092\u4f7f\u3063\u305f\u516c\u5f0f\u3067\u7279\u6b8a\u89e3\u3092\u6c42\u3081\u3066\u307f\u308b\u3002<\/p>\n<p>\u307e\u305a\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f\u306f <code>eq2<\/code> \u306e\u53f3\u8fba\u3092\u30bc\u30ed\u3068\u3057\u305f\u5f0f\u3067\u3042\u308b\u304b\u3089&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[75]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">eq0<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/span><span class=\"n\">eq2<\/span><span class=\"o\">.<\/span><span class=\"n\">lhs<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">eq0<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[75]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle a^{2} y{\\left(x \\right)} + \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)} = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u5fae\u5206\u65b9\u7a0b\u5f0f <code>eq0<\/code> \u3092 <code>y(x)<\/code> \u306b\u3064\u3044\u3066\u89e3\u304f\u3068&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[76]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eq0<\/span><span class=\"p\">,<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[76]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = C_{1} \\sin{\\left(a x \\right)} + C_{2} \\cos{\\left(a x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u4e0a\u8a18\u306e\u3088\u3046\u306b\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e1\u6b21\u72ec\u7acb\u306a\u57fa\u672c\u89e3\u306f\u305d\u308c\u305e\u308c&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[77]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">y1<\/span> <span class=\"o\">=<\/span> <span class=\"n\">cos<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n<span class=\"n\">y2<\/span> <span class=\"o\">=<\/span> <span class=\"n\">sin<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u30ed\u30f3\u30b9\u30ad\u30a2\u30f3\u306f $W(x) \\equiv y_1 y_2&#8242; -y_1&#8242; y_2 = \\cdots$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[78]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">W<\/span> <span class=\"o\">=<\/span> <span class=\"p\">(<\/span><span class=\"n\">y1<\/span><span class=\"o\">*<\/span><span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">y2<\/span><span class=\"p\">,<\/span> <span class=\"n\">x<\/span><span class=\"p\">)<\/span> <span class=\"o\">-<\/span> <span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">y1<\/span><span class=\"p\">,<\/span> <span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"n\">y2<\/span><span class=\"p\">)<\/span><span class=\"o\">.<\/span><span class=\"n\">simplify<\/span><span class=\"p\">()<\/span>\r\n<span class=\"n\">W<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[78]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle a$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3 $y_s(x)$ \u306f\uff0c\u975e\u540c\u6b21\u9805\u3092 $R(x) = \\sin b x$ \u3068\u3057\u3066<\/p>\n<p>$$y_s(x) = y_2(x) \\int^x \\frac{R(x\u2019) y_1 (x\u2019)}{W(x\u2019)} dx\u2019 \u2013y_1(x) \\int^x \\frac{R(x\u2019) y_2(x\u2019)}{W(x\u2019)} dx\u2019$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[79]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'a b'<\/span><span class=\"p\">,<\/span> <span class=\"n\">positive<\/span> <span class=\"o\">=<\/span> <span class=\"kc\">True<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">R<\/span> <span class=\"o\">=<\/span> <span class=\"n\">sin<\/span><span class=\"p\">(<\/span><span class=\"n\">b<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">ys<\/span> <span class=\"o\">=<\/span> <span class=\"p\">(<\/span> <span class=\"n\">y2<\/span> <span class=\"o\">*<\/span> <span class=\"n\">integrate<\/span><span class=\"p\">(<\/span><span class=\"n\">R<\/span><span class=\"o\">*<\/span><span class=\"n\">y1<\/span><span class=\"o\">\/<\/span><span class=\"n\">W<\/span><span class=\"p\">,<\/span> <span class=\"n\">x<\/span><span class=\"p\">)<\/span> \r\n      <span class=\"o\">-<\/span><span class=\"n\">y1<\/span> <span class=\"o\">*<\/span> <span class=\"n\">integrate<\/span><span class=\"p\">(<\/span><span class=\"n\">R<\/span><span class=\"o\">*<\/span><span class=\"n\">y2<\/span><span class=\"o\">\/<\/span><span class=\"n\">W<\/span><span class=\"p\">,<\/span> <span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n\r\n<span class=\"n\">ys<\/span><span class=\"o\">.<\/span><span class=\"n\">simplify<\/span><span class=\"p\">()<\/span><span class=\"o\">.<\/span><span class=\"n\">expand<\/span><span class=\"p\">()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[79]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\begin{cases} \\frac{\\sin{\\left(b x \\right)}}{a^{2} -b^{2}} &amp; \\text{for}\\: a \\neq b \\\\-\\frac{x \\cos{\\left(a x \\right)}}{2 a} + \\frac{\\sin{\\left(b x \\right)} \\cos{\\left(a x -b x \\right)}}{2 a b} &amp; \\text{otherwise} \\end{cases}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u4e0a\u8a18\u306e otherwise \u306e\u5834\u5408\u306e\u7b54\u3048\u304c\u4eca\u3072\u3068\u3064\u3067\u3059\u3002\u76f4\u63a5 $a = b$ \u3068\u3059\u308c\u3070&#8230;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[80]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">R<\/span> <span class=\"o\">=<\/span> <span class=\"n\">sin<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">ys<\/span> <span class=\"o\">=<\/span> <span class=\"p\">(<\/span> <span class=\"n\">y2<\/span> <span class=\"o\">*<\/span> <span class=\"n\">integrate<\/span><span class=\"p\">(<\/span><span class=\"n\">R<\/span><span class=\"o\">*<\/span><span class=\"n\">y1<\/span><span class=\"o\">\/<\/span><span class=\"n\">W<\/span><span class=\"p\">,<\/span> <span class=\"n\">x<\/span><span class=\"p\">)<\/span> \r\n      <span class=\"o\">-<\/span><span class=\"n\">y1<\/span> <span class=\"o\">*<\/span> <span class=\"n\">integrate<\/span><span class=\"p\">(<\/span><span class=\"n\">R<\/span><span class=\"o\">*<\/span><span class=\"n\">y2<\/span><span class=\"o\">\/<\/span><span class=\"n\">W<\/span><span class=\"p\">,<\/span> <span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n\r\n<span class=\"n\">ys<\/span><span class=\"o\">.<\/span><span class=\"n\">simplify<\/span><span class=\"p\">()<\/span><span class=\"o\">.<\/span><span class=\"n\">expand<\/span><span class=\"p\">()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[80]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle -\\frac{x \\cos{\\left(a x \\right)}}{2 a} + \\frac{\\sin{\\left(a x \\right)}}{2 a^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>$a = b$ \u306e\u5834\u5408\u306e\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3\u3092\u6c42\u3081\u305f\u306f\u305a\u306a\u306e\u306b\uff0c$\\sin (a x)$ \u306b\u6bd4\u4f8b\u3059\u308b\u9805\u3082\u51fa\u3066\u304d\u307e\u3057\u305f\u3002\u3053\u308c\u306f\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\u306b\u5438\u53ce\u3067\u304d\u308b\u9805\u3067\u3057\u305f\u306d\u3002<\/p>\n<p>\u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u30ed\u30f3\u30b9\u30ad\u30a2\u30f3\u3092\u4f7f\u3063\u305f\u516c\u5f0f\u3067\u6c42\u3081\u305f\u7279\u6b8a\u89e3\u306f<\/p>\n<p>$\\displaystyle -\\frac{x \\cos (ax)}{2a}$ \u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n<h4 id=\"\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\">\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3<\/h4>\n<p>\u6700\u7d42\u7684\u306b\uff0c\u3053\u306e\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\u306f\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\uff08\u3064\u307e\u308a\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u57fa\u672c\u89e3 $y_1, \\, y_2$ \u306e\u7dda\u5f62\u548c $C_1 y_1 + C_2 y_2$\uff09\u306b\uff0c\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3 $y_s$ \u3092\u8db3\u3057\u305f\u3082\u306e\u3060\u304b\u3089\uff0c<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[81]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\u306f...<\/span>\r\n<span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eq0<\/span><span class=\"p\">,<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span><span class=\"o\">.<\/span><span class=\"n\">rhs<\/span> <span class=\"o\">-<\/span> <span class=\"n\">x<\/span><span class=\"o\">*<\/span><span class=\"n\">cos<\/span><span class=\"p\">(<\/span><span class=\"n\">a<\/span><span class=\"o\">*<\/span><span class=\"n\">x<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span><span class=\"mi\">2<\/span><span class=\"o\">*<\/span><span class=\"n\">a<\/span><span class=\"p\">)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[81]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle C_{1} \\sin{\\left(a x \\right)} + C_{2} \\cos{\\left(a x \\right)} -\\frac{x \\cos{\\left(a x \\right)}}{2 a}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing text_cell rendered\">\n<div class=\"prompt input_prompt\"><\/div>\n<div class=\"inner_cell\">\n<div class=\"text_cell_render border-box-sizing rendered_html\">\n<p>\u6ce8\u610f\uff1a<code>dsolve(eq0, y(x))<\/code> \u306e\u307e\u307e\u3067\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u7b49\u5f0f\u3092\u51fa\u529b\u3059\u308b\u306e\u3067\uff0c\u6b32\u3057\u3044\u306e\u306f\u53f3\u8fba\u3060\u304b\u3089 <code>.rhs<\/code> \u3067\u53f3\u8fba\uff08right-hand-side\uff09\u3092\u53d6\u308a\u51fa\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"cell border-box-sizing code_cell rendered\">\n<div class=\"input\">\n<div class=\"prompt input_prompt\">In\u00a0[82]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">dsolve<\/span><span class=\"p\">(<\/span><span class=\"n\">eq0<\/span><span class=\"p\">,<\/span> <span class=\"n\">y<\/span><span class=\"p\">(<\/span><span class=\"n\">x<\/span><span class=\"p\">))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"output_wrapper\">\n<div class=\"output\">\n<div class=\"output_area\">\n<div class=\"prompt output_prompt\">Out[82]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = C_{1} \\sin{\\left(a x \\right)} + C_{2} \\cos{\\left(a x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":6176,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-6178","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6178","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=6178"}],"version-history":[{"count":9,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6178\/revisions"}],"predecessor-version":[{"id":6273,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6178\/revisions\/6273"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6176"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=6178"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}