{"id":6736,"date":"2023-09-28T14:34:16","date_gmt":"2023-09-28T05:34:16","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=6736"},"modified":"2024-09-26T15:22:53","modified_gmt":"2024-09-26T06:22:53","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\/%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf%e6%bc%94%e7%bf%92\/python-%e3%81%a7%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf%e6%bc%94%e7%bf%92\/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>\u300c\u7406\u5de5\u7cfb\u306e\u6570\u5b66C\u300d\u306e\u300c<a href=\"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\/\" target=\"_blank\" rel=\"noopener\">SymPy \u3067\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/a>\u300d\u306e\u629c\u7c8b\u30d0\u30fc\u30b8\u30e7\u30f3\u3002\u8a73\u7d30\u306f\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n<ul>\n<li><a href=\"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\/\" target=\"_blank\" rel=\"noopener\">\u7406\u5de5\u7cfb\u306e\u6570\u5b66C\uff1aSymPy \u3067\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/a><\/li>\n<\/ul>\n<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=\"\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5-dsolve()\">\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5 <code>dsolve()<\/code><\/h3>\n<p>Python \u3067\u306f\uff0cSymPy \u306e <code>dsolve()<\/code> \u3092\u4f7f\u3063\u30661\u968e\u304a\u3088\u30732\u968e\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\u307e\u305a\u5fc5\u8981\u306a\u30d1\u30c3\u30b1\u30fc\u30b8\u3092 import \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[1]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"kn\">from<\/span> <span class=\"nn\">sympy.abc<\/span> <span class=\"kn\">import<\/span> <span class=\"o\">*<\/span>\r\n<span class=\"kn\">from<\/span> <span class=\"nn\">sympy<\/span> <span class=\"kn\">import<\/span> <span class=\"o\">*<\/span> \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<h4 id=\"1\u968e\u7dda\u5f62\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4f8b\">1\u968e\u7dda\u5f62\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4f8b<\/h4>\n<p>$$ \\frac{dy}{dx} = &#8211; 2 x\\, y $$<\/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[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\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\">$\\displaystyle \\frac{d}{d x} y{\\left(x \\right)} = &#8211; 2 x y{\\left(x \\right)}$<\/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[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<h4 id=\"\u30de\u30eb\u30b5\u30b9\u306e\u4eba\u53e3\u30e2\u30c7\u30eb\">\u30de\u30eb\u30b5\u30b9\u306e\u4eba\u53e3\u30e2\u30c7\u30eb<\/h4>\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[4]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u30ae\u30ea\u30b7\u30e3\u6587\u5b57\u3082 from sympy.abc import * \u3067\u5b9a\u7fa9\u6e08\u307f<\/span>\r\n<span class=\"c1\"># var('gamma')<\/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\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[4]:<\/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[5]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u521d\u671f\u6761\u4ef6\u5909\u6570\u306f2\u6587\u5b57\u5909\u6570\u3060\u304b\u3089\u5b9a\u7fa9\u3057\u3088\u3046<\/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=\"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[5]:<\/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<h5 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<\/h5>\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[6]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">usa<\/span> <span class=\"o\">=<\/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$N_0 = 3.9$ \u3002Python \u306e\u30ea\u30b9\u30c8\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[7]:<\/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>\u6b8b\u308a\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc $\\gamma_1$ \u306f\uff0c\u5225\u306e\u6642\u523b $t_1$ \u306b\u304a\u3051\u308b $N_1 = N_m(t_1)$ \u304b\u3089\u6c42\u3081\u308b\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\nN_1 &amp;=&amp; N_0 e^{\\gamma_1 (t_1 &#8211; t_0)} \\\\<br \/>\n\\log \\frac{N_1}{N_0} &amp;=&amp; \\gamma_1 (t_1 &#8211; t_0) \\\\<br \/>\n\\therefore\\ \\ \\gamma_1 &amp;=&amp; \\frac{1}{t_1 &#8211; t_0} \\log \\frac{N_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[8]:<\/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=\"n\">gamma1<\/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=\"n\">gamma1<\/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_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[9]:<\/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=\"k\">return<\/span> <span class=\"n\">N0<\/span> <span class=\"o\">*<\/span> <span class=\"n\">exp<\/span><span class=\"p\">(<\/span><span class=\"n\">gamma1<\/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 $N(t)$ \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[10]:<\/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 \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6771\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/pmathc101-1.svg\" alt=\"\" width=\"640\" height=\"427\" \/><\/p>\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>SymPy Plotting Backends (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>\u4e00\u65b9\uff0c\u30ea\u30b9\u30c8 <code>usa<\/code> \u306b\u306f<\/p>\n<pre><code>usa = [\r\n [x1, y1],\r\n [x2, y2],\r\n ...,\r\n [xn, yn]\r\n]<\/code><\/pre>\n<p>\u306e\u3088\u3046\u306b\u6570\u5024\u30c7\u30fc\u30bf\u304c\u683c\u7d0d\u3055\u308c\u3066\u3044\u308b\u305f\u3081\uff0c$x$ \u5ea7\u6a19\u306e\u307f\uff0c$y$ \u5ea7\u6a19\u306e\u307f\u306e\u30ea\u30b9\u30c8\u3092\u4f5c\u6210\u3059\u308b\u305f\u3081\u306b\u5c11\u3057\u3060\u3051\u5de5\u592b\u304c\u5fc5\u8981\u3067\u3059\u3002\u3084\u308a\u65b9\u30922\u3064\u307b\u3069&#8230;<\/p>\n<ol>\n<li><code>for<\/code> \u6587\u3092\u4f7f\u3063\u3066\u30ea\u30b9\u30c8\u3092\u4f5c\u6210\u3059\u308b\u4f8b\u3002<\/li>\n<li><code>numpy.array()<\/code> \u3067\u914d\u5217\u306b\u5909\u63db\u3057\u3066\u300c\u5217\u300d\u6210\u5206\u3092\u6307\u5b9a\u3059\u308b\u4f8b\u3002<\/li>\n<\/ol>\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\"># SPB \u30672\u6b21\u5143\u30ea\u30b9\u30c8 usa \u3092 plot_list() \u3059\u308b\u4f8b 1<\/span>\r\n<span class=\"c1\"># for \u6587\u3067 x \u6210\u5206\u30ea\u30b9\u30c8 usaX \u3068 y \u6210\u5206\u30ea\u30b9\u30c8 usaY \u3092\u4f5c\u6210<\/span>\r\n<span class=\"n\">usaX<\/span> <span class=\"o\">=<\/span> <span class=\"p\">[]<\/span>\r\n<span class=\"n\">usaY<\/span> <span class=\"o\">=<\/span> <span class=\"p\">[]<\/span>\r\n<span class=\"k\">for<\/span> <span class=\"n\">dat<\/span> <span class=\"ow\">in<\/span> <span class=\"n\">usa<\/span><span class=\"p\">:<\/span>\r\n    <span class=\"n\">usaX<\/span><span class=\"o\">.<\/span><span class=\"n\">append<\/span><span class=\"p\">(<\/span><span class=\"n\">dat<\/span><span class=\"p\">[<\/span><span class=\"mi\">0<\/span><span class=\"p\">])<\/span>\r\n    <span class=\"n\">usaY<\/span><span class=\"o\">.<\/span><span class=\"n\">append<\/span><span class=\"p\">(<\/span><span class=\"n\">dat<\/span><span class=\"p\">[<\/span><span class=\"mi\">1<\/span><span class=\"p\">])<\/span>\r\n\r\n<span class=\"n\">p2<\/span> <span class=\"o\">=<\/span> <span class=\"n\">plot_list<\/span><span class=\"p\">(<\/span><span class=\"n\">usaX<\/span><span class=\"p\">,<\/span> <span class=\"n\">usaY<\/span><span class=\"p\">,<\/span> <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 \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6772\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/pmathc102-1.svg\" alt=\"\" width=\"640\" height=\"427\" \/><\/p>\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[12]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># SPB \u30672\u6b21\u5143\u30ea\u30b9\u30c8 usa \u3092 plot_list() \u3059\u308b\u4f8b 2<\/span>\r\n<span class=\"c1\"># NumPy \u914d\u5217\u306b\u5909\u63db\u3057\u30661\u5217\u76ee [:,0] 2\u5217\u76ee [:,1] \u3092\u6307\u5b9a<\/span>\r\n<span class=\"kn\">import<\/span> <span class=\"nn\">numpy<\/span> <span class=\"k\">as<\/span> <span class=\"nn\">np<\/span>\r\n<span class=\"n\">npusa<\/span> <span class=\"o\">=<\/span> <span class=\"n\">np<\/span><span class=\"o\">.<\/span><span class=\"n\">array<\/span><span class=\"p\">(<\/span><span class=\"n\">usa<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"n\">p3<\/span> <span class=\"o\">=<\/span> <span class=\"n\">plot_list<\/span><span class=\"p\">(<\/span><span class=\"n\">npusa<\/span><span class=\"p\">[:,<\/span><span class=\"mi\">0<\/span><span class=\"p\">],<\/span> <span class=\"n\">npusa<\/span><span class=\"p\">[:,<\/span><span class=\"mi\">1<\/span><span class=\"p\">],<\/span> <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 \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6773\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/pmathc103-1.svg\" alt=\"\" width=\"640\" height=\"427\" \/><\/p>\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>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\">p4<\/span> <span class=\"o\">=<\/span> <span class=\"n\">p1<\/span> <span class=\"o\">+<\/span> <span class=\"n\">p2<\/span>\r\n<span class=\"n\">p4<\/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 \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6774\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/pmathc104-1.svg\" alt=\"\" width=\"640\" height=\"427\" \/><\/p>\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=\"\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<\/h4>\n<p>$$<br \/>\n\\frac{dN}{dt} = \\gamma N \\left(1 &#8211; \\frac{N}{N_{\\rm max}}\\right)<br \/>\n$$<\/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\">'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\">$\\displaystyle \\frac{d}{d t} N{\\left(t \\right)} = \\gamma \\left(1 &#8211; \\frac{N{\\left(t \\right)}}{Nmax}\\right) N{\\left(t \\right)}$<\/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\">$\\displaystyle N{\\left(t \\right)} = \\frac{N_{0} Nmax e^{\\gamma t}}{N_{0} e^{\\gamma t} &#8211; \\left(N_{0} &#8211; Nmax\\right) e^{\\gamma t_{0}}}$<\/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<h5 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<\/h5>\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<p>$t_1$\uff08\u5e74\uff09\u3068 $t_2$\uff08\u5e74\uff09\u306e\u5024\u3092\u4f7f\u3063\u3066 $\\gamma$ \u3068 $N_{\\rm max}$ \u3092\u6c42\u3081\u307e\u3059\u3002\u6307\u6570\u95a2\u6570\u3092\u542b\u3080\u9023\u7acb\u65b9\u7a0b\u5f0f\u306f\u306a\u304b\u306a\u304b\u89e3\u3044\u3066\u304f\u308c\u306a\u3044\u304b\u3082\u3057\u308c\u306a\u3044\u306e\u3067\uff0c\u7c21\u5358\u306a\u4ee3\u6570\u65b9\u7a0b\u5f0f\u306e\u5f62\u306b\u3057\u3066\u89e3\u304d\u307e\u3059\u3002<\/p>\n<p>$N(t)$ \u306e\u5206\u5b50\u5206\u6bcd\u3092 $e^{\\gamma t}$ \u3067\u5272\u308a\uff0c\u3055\u3089\u306b<\/p>\n<p>\\begin{eqnarray}<br \/>\nn_0 &amp;\\equiv&amp; \\frac{N_0}{N_{\\rm max}}<br \/>\n\\end{eqnarray}<\/p>\n<p>\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<span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'n0 gamma2'<\/span><span class=\"p\">)<\/span>\r\n\r\n<span class=\"k\">def<\/span> <span class=\"nf\">Nv1<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">):<\/span>\r\n    <span class=\"k\">return<\/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=\"p\">(<\/span><span class=\"mi\">1<\/span><span class=\"o\">-<\/span><span class=\"n\">n0<\/span><span class=\"p\">)<\/span><span class=\"o\">*<\/span><span class=\"n\">exp<\/span><span class=\"p\">(<\/span><span class=\"n\">gamma2<\/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\">Nv1<\/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[16]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{3.9}{n_{0} + \\left(1 &#8211; n_{0}\\right) e^{\\gamma_{2} \\cdot \\left(1790 &#8211; t\\right)}}$<\/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>$t_1 = 1850$\uff08\u5e74\uff09\u3068 $t_2 = 1910$\uff08\u5e74\uff09\u306e\u5024\u3092\u5165\u308c\u3066\u9023\u7acb\u65b9\u7a0b\u5f0f\u306e\u5f62\u306b\u3057\u307e\u3059\u3002\u7c21\u5358\u306a\u9023\u7acb\u65b9\u7a0b\u5f0f\u306b\u3059\u308b\u305f\u3081\uff0c\u3055\u3089\u306b $T \\equiv e^{-60\\, \\gamma_2}$ \u3068\u3057\uff0c$n_0$ \u3068 $T$ \u306b\u95a2\u3059\u308b\u30b7\u30f3\u30d7\u30eb\u306a\u9023\u7acb\u65b9\u7a0b\u5f0f\u306e\u5f62\u306b\u3057\u3066\uff0c<code>solve()<\/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[17]:<\/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\">eq1<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/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> <span class=\"n\">Nv1<\/span><span class=\"p\">(<\/span><span class=\"n\">t1<\/span><span class=\"p\">))<\/span> \r\n\r\n<span class=\"c1\"># var('T') \u306f\u7701\u7565\u53ef<\/span>\r\n<span class=\"n\">eq1<\/span> <span class=\"o\">=<\/span> <span class=\"n\">eq1<\/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\">gamma2<\/span><span class=\"p\">),<\/span> <span class=\"n\">T<\/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[17]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle 23.2 = \\frac{3.9}{T \\left(1 &#8211; n_{0}\\right) + n_{0}}$<\/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[18]:<\/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\">eq2<\/span> <span class=\"o\">=<\/span> <span class=\"n\">Eq<\/span><span class=\"p\">(<\/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> <span class=\"n\">Nv1<\/span><span class=\"p\">(<\/span><span class=\"n\">t2<\/span><span class=\"p\">))<\/span>\r\n<span class=\"n\">eq2<\/span> <span class=\"o\">=<\/span> <span class=\"n\">eq2<\/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\">gamma2<\/span><span class=\"p\">),<\/span> <span class=\"n\">T<\/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[18]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle 92.0 = \\frac{3.9}{T^{2} \\cdot \\left(1 &#8211; n_{0}\\right) + n_{0}}$<\/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[19]:<\/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\">eq1<\/span><span class=\"p\">,<\/span> <span class=\"n\">eq2<\/span><span class=\"p\">],<\/span> <span class=\"p\">[<\/span><span class=\"n\">n0<\/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[19]:<\/div>\n<div class=\"output_text output_subarea output_execute_result\">\n<pre>[(0.0200125277046207, 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>$n_0, \\ T$ \u304b\u3089\u3082\u3068\u306e $N_{\\rm max}, \\ \\gamma_2$ \u306e\u5024\u306b\u306a\u304a\u3059\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[20]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">n0<\/span><span class=\"p\">,<\/span> <span class=\"n\">T<\/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<span class=\"n\">Nmax<\/span> <span class=\"o\">=<\/span> <span class=\"n\">N0<\/span><span class=\"o\">\/<\/span><span class=\"n\">n0<\/span>\r\n<span class=\"n\">gamma2<\/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\">T<\/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=\"k\">return<\/span> <span class=\"n\">N0<\/span><span class=\"o\">*<\/span><span class=\"n\">Nmax<\/span><span class=\"o\">*<\/span><span class=\"n\">exp<\/span><span class=\"p\">(<\/span><span class=\"n\">gamma2<\/span><span class=\"o\">*<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/span><span class=\"o\">\/<\/span><span class=\"p\">(<\/span>\r\n        <span class=\"n\">N0<\/span><span class=\"o\">*<\/span><span class=\"n\">exp<\/span><span class=\"p\">(<\/span><span class=\"n\">gamma2<\/span><span class=\"o\">*<\/span><span class=\"n\">t<\/span><span class=\"p\">)<\/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\">gamma2<\/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 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\">p5<\/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 \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6775\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/pmathc105-1.svg\" alt=\"\" width=\"640\" height=\"427\" \/><\/p>\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[22]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"n\">p6<\/span> <span class=\"o\">=<\/span> <span class=\"n\">p1<\/span><span class=\"o\">+<\/span><span class=\"n\">p2<\/span><span class=\"o\">+<\/span><span class=\"n\">p5<\/span>\r\n<span class=\"n\">p6<\/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 \">\n<p><!--?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?--><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6776\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/pmathc106-1.svg\" alt=\"\" width=\"640\" height=\"427\" \/><\/p>\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=\"\u6700\u3082\u7c21\u5358\u306a2\u968e\u7dda\u5f62\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4f8b\">\u6700\u3082\u7c21\u5358\u306a2\u968e\u7dda\u5f62\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4f8b<\/h4>\n<p>$y^{\\prime\\prime} + K y = 0$ \u3042\u308b\u3044\u306f\u79fb\u9805\u3057\u3066 $y^{\\prime\\prime} = &#8211; K y$ \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<h5 id=\"$K-&gt;-0$-\u306e\u5834\u5408\">$K &gt; 0$ \u306e\u5834\u5408<\/h5>\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\">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=\"c1\"># var('x')<\/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[23]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)} = &#8211; K y{\\left(x \\right)}$<\/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[24]:<\/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[24]:<\/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<h5 id=\"$K-&lt;-0$-\u306e\u5834\u5408\">$K &lt; 0$ \u306e\u5834\u5408<\/h5>\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\">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[25]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)} = &#8211; K y{\\left(x \\right)}$<\/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[26]:<\/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[26]:<\/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<h5 id=\"$K-=-0$-\u306e\u5834\u5408\">$K = 0$ \u306e\u5834\u5408<\/h5>\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\">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[27]:<\/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[28]:<\/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[28]:<\/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<h4 id=\"\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\">\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h4>\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\u4e00\u822c\u89e3 $y$ \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5834\u5408\u5206\u3051\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[29]:<\/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=\"c1\"># var('x')<\/span>\r\n<span class=\"c1\"># var('b c')<\/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[29]:<\/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[30]:<\/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[30]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = \\left(C_{1} e^{x \\sqrt{b^{2} &#8211; c}} + C_{2} e^{- x \\sqrt{b^{2} &#8211; c}}\\right) e^{- b x}$<\/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=\"\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4f8b\">\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4f8b<\/h4>\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<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[31]:<\/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[31]:<\/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[32]:<\/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[32]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = C_{1} \\sin{\\left(a x \\right)} + C_{2} \\cos{\\left(a x \\right)} + \\frac{\\sin{\\left(b x \\right)}}{a^{2} &#8211; b^{2}}$<\/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[33]:<\/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[33]:<\/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[34]:<\/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[34]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(x \\right)} = C_{2} \\sin{\\left(a x \\right)} + \\left(C_{1} &#8211; \\frac{x}{2 a}\\right) \\cos{\\left(a x \\right)}$<\/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=\"\u7df4\u7fd2-2-1\">\u7df4\u7fd2 2-1<\/h4>\n<p><strong>\u91cd\u529b\u5834\u4e2d\u306e\u6295\u3051\u3099\u4e0a\u3051\u3099\u904b\u52d5<\/strong><\/p>\n<ol>\n<li>\u9ad8\u3055 $0$ \u304b\u3089\u521d\u901f\u5ea6 $v_0$ \u3066\u3099\u925b\u76f4\u4e0a\u65b9\u306b\u6295\u3051\u3099\u4e0a\u3051\u3099\u305f\u7269\u4f53\u306e\u6642\u523b $t$ \u306b\u304a\u3051\u308b\u9ad8\u3055 $y$ \u3092\u6c42\u3081\u306a\u3055\u3044\u3002\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f\uff0c<\/li>\n<\/ol>\n<p>$$ \\frac{d^2 y}{dt^2} = -g$$<\/p>\n<ol>\n<li>\u901f\u5ea6\u306b\u6bd4\u4f8b\u3059\u308b\u7a7a\u6c17\u62b5\u6297\u304b\u3099\u3042\u308b\u5834\u5408\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3002<br \/>\n\u3053\u306e\u3068\u304d\uff0c\u9ad8\u3055 $0$ \u304b\u3089\u521d\u901f\u5ea6 $v_0$ \u3066\u3099\u925b\u76f4\u4e0a\u65b9\u306b\u6295\u3051\u3099\u4e0a\u3051\u3099\u305f\u7269\u4f53\u306e\u6642\u523b $t$ \u306b\u304a\u3051\u308b\u9ad8\u3055 $y$ \u3092<br \/>\n\u6c42\u3081\u306a\u3055\u3044\u3002<\/li>\n<\/ol>\n<p>$$ \\frac{d^2 y}{dt^2} = -g &#8211; \\beta \\frac{dy}{dt}$$<\/p>\n<ol>\n<li>\u524d\u554f 2. \u3066\u3099\u6c42\u3081\u305f\u89e3\u304b\u3099 $\\beta \\rightarrow 0$ \u306e\u3068\u304d\uff0c\u524d\u554f 1. \u306e\u7b54\u3048\u306b\u4e00\u81f4\u3059\u308b\u3053\u3068\u3092\u793a\u3057\u306a\u3055\u3044\u3002<\/li>\n<\/ol>\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=\"c1\"># 1. <\/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<span class=\"c1\"># var('t')<\/span>\r\n<span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'g'<\/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\">t<\/span><span class=\"p\">),<\/span> <span class=\"n\">t<\/span><span class=\"p\">,<\/span> <span class=\"mi\">2<\/span><span class=\"p\">),<\/span> <span class=\"o\">-<\/span><span class=\"n\">g<\/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[35]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle \\frac{d^{2}}{d t^{2}} y{\\left(t \\right)} = &#8211; g$<\/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\">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\">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[36]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle y{\\left(t \\right)} = C_{1} + C_{2} t &#8211; \\frac{g t^{2}}{2}$<\/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[37]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># \u521d\u671f\u6761\u4ef6\u3092\u4e0e\u3048\u3066\u89e3\u304f\u4f8b<\/span>\r\n<span class=\"c1\"># 1\u6587\u5b57\u5909\u6570\u4ee5\u5916\u3092\u4f7f\u3046\u3068\u304d\u306f var() \u3067\u5ba3\u8a00\u3057\u3066\u304a\u304f<\/span>\r\n<span class=\"n\">var<\/span><span class=\"p\">(<\/span><span class=\"s1\">'v0'<\/span><span class=\"p\">)<\/span>\r\n\r\n<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\">t<\/span><span class=\"p\">),<\/span> \r\n       <span class=\"n\">ics<\/span> <span class=\"o\">=<\/span> <span class=\"p\">{<\/span><span class=\"n\">y<\/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=\"mi\">0<\/span><span class=\"p\">):<\/span><span class=\"mi\">0<\/span><span class=\"p\">,<\/span>          <span class=\"c1\"># y(0) = 0<\/span>\r\n              <span class=\"n\">diff<\/span><span class=\"p\">(<\/span><span class=\"n\">y<\/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=\"o\">.<\/span><span class=\"n\">subs<\/span><span class=\"p\">(<\/span><span class=\"n\">t<\/span><span class=\"p\">,<\/span> <span class=\"mi\">0<\/span><span class=\"p\">):<\/span><span class=\"n\">v0<\/span> <span class=\"c1\"># dy\/dt(0) = v0<\/span>\r\n             <span class=\"p\">}<\/span>\r\n      <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\">$\\displaystyle y{\\left(t \\right)} = &#8211; \\frac{g t^{2}}{2} + t v_{0}$<\/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[38]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># 2. \u3082\u540c\u69d8\u306b<\/span>\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[39]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre><span class=\"c1\"># 3. \r\n<\/span># \u7b49\u5f0f eq \u306e\u53f3\u8fba\u306f eq.rhs\r\n<span class=\"c1\"># \u6975\u9650\u306e\u4f8b<\/span>\r\n\r\n<span class=\"n\">limit<\/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> <span class=\"n\">x<\/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[39]:<\/div>\n<div class=\"output_latex output_subarea output_execute_result\">$\\displaystyle 1$<\/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[\u00a0]:<\/div>\n<div class=\"inner_cell\">\n<div class=\"input_area\">\n<div class=\" highlight hl-ipython3\">\n<pre> \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u300c\u7406\u5de5\u7cfb\u306e\u6570\u5b66C\u300d\u306e\u300cSymPy \u3067\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u300d\u306e\u629c\u7c8b\u30d0\u30fc\u30b8\u30e7\u30f3\u3002\u8a73\u7d30\u306f\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p><p><a class=\"more-link btn\" href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf%e6%bc%94%e7%bf%92\/python-%e3%81%a7%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf%e6%bc%94%e7%bf%92\/sympy-%e3%81%a7%e5%b8%b8%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%bc%8f\/\">\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n<ul>\n<li>\u7406\u5de5\u7cfb\u306e\u6570\u5b66C\uff1aSymPy \u3067\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/li>\n<\/ul>\n","protected":false},"author":33,"featured_media":0,"parent":6721,"menu_order":20,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-6736","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\/6736","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=6736"}],"version-history":[{"count":8,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6736\/revisions"}],"predecessor-version":[{"id":9431,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6736\/revisions\/9431"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6721"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=6736"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}