{"id":4498,"date":"2024-12-18T16:00:45","date_gmt":"2024-12-18T07:00:45","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=4498"},"modified":"2024-12-18T21:51:48","modified_gmt":"2024-12-18T12:51:48","slug":"sympy-%e3%81%a7%ef%bc%88%e3%81%82%e3%81%88%e3%81%a6%ef%bc%89%e6%95%b0%e5%80%a4%e8%a7%a3%e6%9e%90","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%e6%95%b0%e5%80%a4%e8%a7%a3%e6%9e%90\/sympy-%e3%81%a7%ef%bc%88%e3%81%82%e3%81%88%e3%81%a6%ef%bc%89%e6%95%b0%e5%80%a4%e8%a7%a3%e6%9e%90\/","title":{"rendered":"SymPy \u3066\u3099\uff08\u3042\u3048\u3066\uff09\u6570\u5024\u89e3\u6790"},"content":{"rendered":"
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Python \u3067\u6570\u5024\u89e3\u6790\u306e\u305f\u3081\u306b\u5fc5\u8981\u306a\u30d7\u30ed\u30b0\u30e9\u30df\u30f3\u30b0\u3068\uff0c\u3044\u304f\u3064\u304b\u306e\u4f8b\u3092\u793a\u3057\u307e\u3059\u3002
\n\u3053\u3053\u3067\u306f\u6570\u5024\u89e3\u6790\u30e9\u30a4\u30d6\u30e9\u30ea\u3067\u3042\u308b SciPy \u3092\u4f7f\u308f\u305a\uff0c\u8a08\u7b97\u6a5f\u4ee3\u6570\u30b7\u30b9\u30c6\u30e0\u3067\u3042\u308b SymPy \u3092\u3042\u3048\u3066\u4f7f\u3063\u305f\u4f8b\u3092\u793a\u3057\u307e\u3059\u3002<\/p>\n

\u307e\u305f SymPy \u81ea\u4f53\u3067\u3082\u30b0\u30e9\u30d5\u4f5c\u6210\u6a5f\u80fd\u304c\u3042\u308a\u307e\u3059\u304c\uff0c\u3053\u3053\u3067\u306f\uff0cSymPy Plotting Backends (SPB)<\/a> \u3092\u4f7f\u3063\u3066\u30b0\u30e9\u30d5\u3092\u63cf\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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\u30e9\u30a4\u30d6\u30e9\u30ea\u306e import<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[1]:<\/div>\n
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# SymPy \u3092\u4f7f\u3046\u3068\u304d\u306e\u304a\u307e\u3058\u306a\u3044<\/span>\r\nfrom<\/span> sympy.abc<\/span> import<\/span> *<\/span>\r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n\r\n# \u30b0\u30e9\u30d5\u306f SymPy Plotting Backends (SPB) \u3067\u63cf\u304f<\/span>\r\nfrom<\/span> spb<\/span> import<\/span> *<\/span>\r\n\r\n# \u4ee5\u4e0b\u306f\u30b0\u30e9\u30d5\u3092 SVG \u3067 Notebook \u306b\u30a4\u30f3\u30e9\u30a4\u30f3\u8868\u793a\u3055\u305b\u308b\u8a2d\u5b9a<\/span>\r\n%<\/span>config<\/span> InlineBackend.figure_formats = ['svg']\r\n\r\n# \u30c7\u30d5\u30a9\u30eb\u30c8\u8a2d\u5b9a\u306e\u305f\u3081<\/span>\r\nimport<\/span> matplotlib.pyplot<\/span> as<\/span> plt<\/span>\r\n# mathtext font \u306e\u8a2d\u5b9a<\/span>\r\nplt<\/span>.<\/span>rcParams<\/span>[<\/span>'mathtext.fontset'<\/span>]<\/span> =<\/span> 'cm'<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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SymPy \u306b\u3088\u308b\u6570\u5024\u5fae\u5206<\/h3>\n
\u89e3\u6790\u7684\u306a\u5fae\u5206\u306e\u5b9a\u7fa9\u306f\uff08\u524d\u65b9\u5dee\u5206\u306e\u6975\u9650\u3068\u3057\u3066\uff09<\/p>\n
$$\\frac{df}{dx} \\equiv \\lim_{h \\rightarrow 0} \\frac{f(x+h) -f(x)}{h}$$<\/p>\n
\u3067\u3057\u305f\u3002\u6ed1\u3089\u304b\u306a\u95a2\u6570\u3067\u3042\u308c\u3070\uff0c\uff08\u5f8c\u65b9\u5dee\u5206\u306e\u6975\u9650\u3068\u3057\u3066\uff09
\n$$\\frac{df}{dx} = \\lim_{h \\rightarrow 0} \\frac{f(x) -f(x-h)}{h}$$
\n\u3068\u3057\u3066\u3082\u826f\u3044\u3067\u3059\u3002<\/p>\n
\u73fe\u5b9f\u4e16\u754c\u3067\u306f $ h\\rightarrow 0$ \u306e\u6975\u9650\u306f\u3068\u308c\u307e\u305b\u3093\u304b\u3089\u5341\u5206\u5c0f\u3055\u3044\u5024\u3068\u3057\u3066 h \u3092\u5b9a\u7fa9\u3057\uff0c\u8fd1\u4f3c\u7684\u306a\u6570\u5024\u5fae\u5206\u3092\uff08\u524d\u65b9\u5dee\u5206\u3068\u5f8c\u65b9\u5dee\u5206\u306e\u5e73\u5747\u3067\u3042\u308b\u4e2d\u5fc3\u5dee\u5206\u3068\u3057\u3066\uff09\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n
\\begin{eqnarray}
\n\\frac{df}{dx} &\\simeq& \\frac{1}{2} \\left\\{\\frac{f(x+h) -f(x)}{h} + \\frac{f(x) -f(x-h)}{h} \\right\\} \\\\
\n&=& \\frac{f(x+h) -f(x-h)}{2h}
\n\\end{eqnarray}<\/p>\n
\u95a2\u6570 $f(x)$ \u304c\u521d\u7b49\u95a2\u6570\uff08\u304a\u3088\u3073\u521d\u7b49\u95a2\u6570\u306e\u5408\u6210\u95a2\u6570\uff09\u3067\u3042\u308b\u5834\u5408\u306f\u89e3\u6790\u7684\u306b\u5fae\u5206\u3067\u304d\u307e\u3059\u304c\uff0c\u305d\u3046\u3067\u306a\u3044\u5834\u5408\uff0c\u4f8b\u3048\u3070 $f(x)$ \u304c\u6570\u5024\u7684\u306b\u3057\u304b\u4e0e\u3048\u3089\u308c\u3066\u3044\u306a\u3044\u3068\u304b\uff0c\u4f55\u304b\u89e3\u6790\u7684\u306b\u5fae\u5206\u3067\u304d\u306a\u3044\u5834\u5408\u306b\u306f\uff0c\u6570\u5024\u5fae\u5206\u306b\u3088\u3063\u3066\u8fd1\u4f3c\u7684\u306a\u5024\u3092\u6c42\u3081\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n
\u307e\u305f\uff0c\u89e3\u6790\u7684\u306b\u5fae\u5206\u3067\u304d\u305d\u3046\u3060\u304c\uff0c\u4eba\u529b\u3067\u306f\u3051\u3063\u3053\u3046\u5927\u5909\u305d\u3046\u306a\u5834\u5408\u306b\u3082\uff0c\u6570\u5024\u5fae\u5206\u306b\u3088\u3063\u3066\u8fd1\u4f3c\u7684\u306a\u5024\u3092\u4f7f\u3046\u307b\u3046\u304c\u4fbf\u5229\u306a\u5834\u5408\u304c\u3042\u308b\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3002<\/p>\n
\u4ee5\u4e0b\u306e\u3088\u3046\u306b\uff0c\u95a2\u6570 f(x)<\/code> \u3092\u6570\u5024\u5fae\u5206\u3057\u3066\u304f\u308c\u308b\u95a2\u6570 ndiff()<\/code> \u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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def<\/span> ndiff<\/span>(<\/span>f<\/span>,<\/span> x<\/span>,<\/span> h<\/span>=<\/span>1e-5<\/span>):<\/span>\r\n    # h \u3092\u7701\u7565\u3059\u308b\u3068 h=1e-5 \u3068\u3057\u3066\u8a08\u7b97\u3059\u308b<\/span>\r\n    return<\/span> (<\/span>f<\/span>(<\/span>x<\/span> +<\/span> h<\/span>)<\/span> -<\/span> f<\/span>(<\/span>x<\/span> -<\/span> h<\/span>))<\/span>\/<\/span>(<\/span>2<\/span>*<\/span>h<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4f8b\u3068\u3057\u3066\uff0c$f(x) = \\sin x$ \u306e $x = 3$ \u306b\u304a\u3051\u308b\u6570\u5024\u5fae\u5206\u3092\u6c42\u3081\u3066\u307f\u307e\u3059\u3002\uff08\u7b54\u3048\u306f $\\cos 3$ \u3067\u3059\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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def<\/span> f<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> sin<\/span>(<\/span>x<\/span>)<\/span>\r\n\r\nfor<\/span> h<\/span> in<\/span> [<\/span>1e-4<\/span>,<\/span> 1e-5<\/span>,<\/span> 1e-6<\/span>,<\/span> 1e-7<\/span>,<\/span> 1e-8<\/span>]:<\/span>\r\n    print<\/span>(<\/span>'h = <\/span>%3.1e<\/span>, ndiff = <\/span>%.15f<\/span>'<\/span> %<\/span> (<\/span>h<\/span>,<\/span> ndiff<\/span>(<\/span>f<\/span>,<\/span> 3<\/span>,<\/span> h<\/span>)))<\/span>\r\n\r\nprint<\/span>(<\/span>' '<\/span>*<\/span>10<\/span> +<\/span> 'cos(<\/span>%3.1f<\/span>) = <\/span>%.15f<\/span>'<\/span> %<\/span> (<\/span>3<\/span>,<\/span> cos<\/span>(<\/span>3<\/span>)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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h = 1.0e-04, ndiff = -0.989992494952602\r\nh = 1.0e-05, ndiff = -0.989992496590319\r\nh = 1.0e-06, ndiff = -0.989992496730485\r\nh = 1.0e-07, ndiff = -0.989992494926373\r\nh = 1.0e-08, ndiff = -0.989992490763036\r\n          cos(3.0) = -0.989992496600445\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4ee5\u4e0a\u306e\u7d50\u679c\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\uff0ch<\/code> \u3092\u3084\u307f\u304f\u3082\u306b\u5c0f\u3055\u304f\u3059\u308c\u3070\u3059\u308b\u307b\u3069\uff0c\u6570\u5024\u5fae\u5206\u306e\u7cbe\u5ea6\u304c\u3042\u304c\u308b\uff0c\u3068\u3044\u3046\u308f\u3051\u3067\u306f\u306a\u3044\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u25cb\u7df4\u7fd2\uff1a\u95a2\u6570\u306e\u6570\u5024\u5fae\u5206<\/h4>\n
$\\displaystyle f(x) = \\frac{x^3}{e^x -1}$ \u306e\u6975\u5927\u5024\u3092\u6c42\u3081\u308b\u305f\u3081\u306e\u6e96\u5099\u3068\u3057\u3066\uff0c\u307e\u305a\uff0c$f(x)$ \u3068\u305d\u306e\u6570\u5024\u5fae\u5206 ndiff(f, x)<\/code> \u3092 $0 \\leq x \\leq 5$ \u306e\u7bc4\u56f2\u3067\u30b0\u30e9\u30d5\u306b\u3059\u308b\u3002<\/p>\n
\uff08$f(x)$ \u306e\u5fae\u5206\u306f\u7c21\u5358\u306b\u6c42\u3081\u3089\u308c\u308b\u306e\u3060\u304c\uff0c\u3053\u3053\u306f\u305b\u3063\u304b\u304f\u899a\u3048\u305f\u306e\u3067\u6570\u5024\u5fae\u5206\u3057\u3066\u307f\u307e\u3059\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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def<\/span> f<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> x<\/span>**<\/span>3<\/span>\/<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>)<\/span> -<\/span> 1<\/span>)<\/span>\r\n\r\ngraphics<\/span>(<\/span>\r\n    line<\/span>(<\/span>f<\/span>(<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> 0.001<\/span>,<\/span> 5<\/span>),<\/span> '$f(x)$'<\/span>),<\/span>\r\n    line<\/span>(<\/span>ndiff<\/span>(<\/span>f<\/span>,<\/span> x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> 0.001<\/span>,<\/span> 5<\/span>),<\/span> '$f^{\\prime}(x)$'<\/span>),<\/span> \r\n    xlabel<\/span> =<\/span> '$x$'<\/span>,<\/span> ylabel<\/span> =<\/span> '\u3000'<\/span>\r\n)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e0a\u306e\u30b0\u30e9\u30d5\u304b\u3089\uff0c\u6570\u5024\u5fae\u5206\u3057\u305f\u5fae\u5206\u4fc2\u6570 $f^{\\prime}(x)$ \u306e\u5024\u304c\u30bc\u30ed\u306b\u306a\u308b $x$ \u306f\uff0c$2 < x < 3$ \u306e\u3042\u305f\u308a\u3067\u3042\u308b\u3053\u3068\uff0c\u305d\u306e\u3042\u305f\u308a\u3067\u78ba\u304b\u306b $f(x)$ \u304c\u6975\u5927\uff08\u6700\u5927\uff09\u3068\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u898b\u3066\u3068\u308c\u307e\u3059\u3002<\/p>\n
\u5ff5\u306e\u305f\u3081\u306b\uff0c\u89e3\u6790\u7684\u306b\u5fae\u5206\u3057\u305f\u7d50\u679c\u3068\u6bd4\u8f03\u3057\u307e\u3059\u3002\u76ee\u8996\u3067\u306f\u307b\u307c\u533a\u5225\u304c\u3064\u304b\u306a\u3044\u3067\u3059\u306d\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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df<\/span> =<\/span> diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>\r\n\r\ngraphics<\/span>(<\/span>\r\n    line<\/span>(<\/span>ndiff<\/span>(<\/span>f<\/span>,<\/span> x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> 0.0001<\/span>,<\/span> 5<\/span>),<\/span> '\u6570\u5024\u5fae\u5206'<\/span>,<\/span>\r\n         rendering_kw<\/span>=<\/span>{<\/span>'lw'<\/span>:<\/span>2<\/span>}),<\/span>\r\n    line<\/span>(<\/span>df<\/span>,<\/span> (<\/span>x<\/span>,<\/span> 0.0001<\/span>,<\/span> 5<\/span>),<\/span> '\u89e3\u6790\u7684\u5fae\u5206'<\/span>,<\/span>\r\n         rendering_kw<\/span>=<\/span>{<\/span>'lw'<\/span>:<\/span>0.5<\/span>})<\/span>\r\n)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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SymPy \u306b\u3088\u308b\u6570\u5024\u7a4d\u5206\uff1aintegrate().evalf()<\/code><\/h3>\n
$\\displaystyle \\int_0^{\\pi} \\sin(\\sin x)\\, dx$ \u306e\u8a08\u7b97\u3092\u3057\u307e\u3059\u3002\u89e3\u6790\u7684\u306b\u89e3\u3051\u306a\u3044\u3068\uff0cSymPy \u306f\u305d\u306e\u307e\u307e\u306e\u5f62\u3067\u8fd4\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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integrate<\/span>(<\/span>sin<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>)),<\/span> (<\/span>x<\/span>,<\/span> 0<\/span>,<\/span> pi<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
$\\displaystyle \\int\\limits_{0}^{\\pi} \\sin{\\left(\\sin{\\left(x \\right)} \\right)}\\, dx$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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.evalf()<\/code> \u3059\u308b\u3068\u6570\u5024\u7a4d\u5206\u3057\u305f\u5024\u3092\u8fd4\u3057\u307e\u3059\u3002\uff08SymPy \u304c\u6570\u5024\u7a4d\u5206\u3057\u3066\u304f\u308c\u308b\u306e\u3067\uff0c\u308f\u3056\u308f\u3056\u81ea\u5206\u3067\u5b9a\u7fa9\u3059\u308b\u5fc5\u8981\u306f\u3042\u308a\u307e\u305b\u3093\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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integrate<\/span>(<\/span>sin<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>)),<\/span> (<\/span>x<\/span>,<\/span> 0<\/span>,<\/span> pi<\/span>))<\/span>.<\/span>evalf<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
$\\displaystyle 1.78648748195005$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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.evalf()<\/code> \u306e\u304b\u308f\u308a\u306b .n()<\/code> \u3092\u4f7f\u3063\u3066\u3082\u3088\u3044\u3088\u3046\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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integrate<\/span>(<\/span>sin<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>)),<\/span> (<\/span>x<\/span>,<\/span> 0<\/span>,<\/span> pi<\/span>))<\/span>.<\/span>n<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
$\\displaystyle 1.78648748195005$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4ee5\u4e0b\u306e\u3088\u3046\u306b N()<\/code> \u3092\u4f7f\u3063\u3066\u3082\u3088\u3044\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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N<\/span>(<\/span>integrate<\/span>(<\/span>sin<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>)),<\/span> (<\/span>x<\/span>,<\/span> 0<\/span>,<\/span> pi<\/span>)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
$\\displaystyle 1.78648748195005$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\n
\u53c2\u8003\uff1a\u30b7\u30f3\u30d7\u30bd\u30f3\u6cd5\u306b\u3088\u308b\u6570\u5024\u7a4d\u5206<\/h4>\n
\u30b7\u30f3\u30d7\u30bd\u30f3\u6cd5\u3067 $\\displaystyle \\int_a^b f(x)\\, dx$ \u3092\u6c42\u3081\u308b\u3002<\/p>\n
\u7a4d\u5206\u533a\u9593 $[a, b]$ \u3092 $N (=2n)$ \u7b49\u5206\uff08\u5076\u6570\u7b49\u5206\uff09\u3057\uff0c<\/p>\n
$$h = \\frac{b-a}{N}, \\quad f_i = f(a + i\\,h)$$<\/p>\n
\u3068\u3059\u308b\u3002\u30b7\u30f3\u30d7\u30bd\u30f3\u6cd5\u306e\u516c\u5f0f\u306f<\/p>\n
\\begin{eqnarray}
\n\\int_a^b f(x)\\, dx &\\simeq&
\n\\frac{h}{3}( f_0 + 4 f_1 + f_2) + \\frac{h}{3}( f_2 + 4 f_3 + f_4) + \\\\
\n&& \\frac{h}{3}( f_4 + 4 f_5 + f_6) + \\frac{h}{3}( f_6 + 4 f_7 + f_8)+ \\cdots\\\\
\n&&+ \\frac{h}{3}( f_{N-2} + 4 f_{N-1} + f_N)\\\\
\n&=&\\frac{h}{3}
\n\\left(f_0 + 2 \\sum_{i=1}^{n-1} f_{2i} + 4 \\sum_{i=1}^{n} f_{2i-1}+ f_{2n}\\right)
\n\\end{eqnarray}<\/p>\n
SymPy \u3067\u306f .evalf()<\/code> \u3092\u4f7f\u3063\u3066\u6570\u5024\u7a4d\u5206\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\u308f\u3056\u308f\u3056\u81ea\u4f5c\u306e\u95a2\u6570\u3092\u3064\u304f\u308b\u5fc5\u8981\u306f\u306a\u3044\u306e\u3067\u3059\u304c\uff0c\u6559\u80b2\u7684\u898b\u5730\u304b\u3089\u3042\u3048\u3066\uff0c\u30b7\u30f3\u30d7\u30bd\u30f3\u6cd5\u306e\u516c\u5f0f\u3092\u4f7f\u3063\u3066\u6570\u5024\u7a4d\u5206\u306e\u5024\u3092\u8fd4\u3059\u95a2\u6570 simpson()<\/code> \u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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def<\/span> simpson<\/span>(<\/span>f<\/span>,<\/span> a<\/span>,<\/span> b<\/span>,<\/span> N<\/span>):<\/span>\r\n    h<\/span> =<\/span> float<\/span>((<\/span>b<\/span> -<\/span> a<\/span>)<\/span>\/<\/span>N<\/span>)<\/span>\r\n    n<\/span> =<\/span> Rational<\/span>(<\/span>N<\/span>,<\/span> 2<\/span>)<\/span>\r\n    tmp<\/span> =<\/span> h<\/span>\/<\/span>3<\/span> *<\/span> (<\/span>f<\/span>(<\/span>a<\/span>)<\/span> +<\/span> \r\n                 2<\/span>*<\/span>summation<\/span>(<\/span>f<\/span>(<\/span>a<\/span> +<\/span> 2<\/span>*<\/span>i<\/span> *<\/span> h<\/span>),<\/span> (<\/span>i<\/span>,<\/span> 1<\/span>,<\/span> n<\/span>-<\/span>1<\/span>))<\/span> +<\/span>\r\n                 4<\/span>*<\/span>summation<\/span>(<\/span>f<\/span>(<\/span>a<\/span> +<\/span> (<\/span>2<\/span>*<\/span>i<\/span>-<\/span>1<\/span>)<\/span> *<\/span> h<\/span>),<\/span> (<\/span>i<\/span>,<\/span> 1<\/span>,<\/span> n<\/span>))<\/span> +<\/span>\r\n                 f<\/span>(<\/span>b<\/span>)<\/span>\r\n                 )<\/span>\r\n    return<\/span> tmp<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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N<\/code> \u306e\u5024\u3092\u5909\u3048\u3066 $\\displaystyle \\int_0^{\\pi} \\sin(\\sin x)\\, dx$ \u3092 simpson()<\/code> \u3067\u6570\u5024\u7a4d\u5206\u3057\u3066\uff0c\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3057\u3066\u307f\u307e\u3059\u3002\uff08N<\/code> \u304c\u5927\u304d\u3044\u5834\u5408\u306f\u6642\u9593\u304c\u304b\u304b\u308a\u307e\u3059\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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def<\/span> f<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> sin<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[12]:<\/div>\n
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for<\/span> N<\/span> in<\/span> [<\/span>10<\/span>*<\/span>2<\/span>**<\/span>i<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>9<\/span>)]:<\/span>\r\n    print<\/span>(<\/span>'N = <\/span>%4d<\/span>, <\/span>%.15f<\/span>'<\/span> %<\/span>(<\/span>N<\/span>,<\/span> simpson<\/span>(<\/span>f<\/span>,<\/span> 0<\/span>,<\/span> pi<\/span>,<\/span> N<\/span>)))<\/span>\r\n\r\nninteg<\/span> =<\/span> integrate<\/span>(<\/span>f<\/span>(<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> 0<\/span>,<\/span> pi<\/span>))<\/span>.<\/span>evalf<\/span>()<\/span>\r\nprint<\/span>(<\/span>'.evalf()  <\/span>%.15f<\/span>'<\/span> %<\/span> ninteg<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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N =   10, 1.786721213744078\r\nN =   20, 1.786501256275661\r\nN =   40, 1.786488331266679\r\nN =   80, 1.786487534856180\r\nN =  160, 1.786487485253951\r\nN =  320, 1.786487482156504\r\nN =  640, 1.786487481962953\r\nN = 1280, 1.786487481950860\r\nN = 2560, 1.786487481950103\r\n.evalf()  1.786487481950052\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u4ee5\u4e0a\u306e\u7d50\u679c\u304b\u3089\uff0c\u3069\u3093\u306a\u3053\u3068\u304c\u308f\u304b\u308b\u304b\u306a\uff1f<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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SymPy \u306b\u3088\u308b\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\uff1ansolve()<\/code><\/h3>\n
SymPy \u306b\u306f\u6570\u5024\u89e3\u3092\u6c42\u3081\u308b\u95a2\u6570 nsolve()<\/code> \u304c\u7528\u610f\u3055\u308c\u3066\u3044\u307e\u3059\u306e\u3067\uff0c\u3053\u308c\u3092\u4f7f\u3063\u3066\u307f\u307e\u3059\u3002\uff08\u308f\u3056\u308f\u3056\u81ea\u5206\u3067\u5b9a\u7fa9\u3059\u308b\u5fc5\u8981\u306f\u3042\u308a\u307e\u305b\u3093\u3002\uff09<\/p>\n
\u307e\u305a\uff0c\u89e3\u3092\u6c42\u3081\u308b\u95a2\u6570 $f(x)$ \u3092\u5b9a\u7fa9\u3057\uff0c$f(x) = 0$ \u3068\u306a\u308b $x$ \u306e\u3042\u305f\u308a\u3092\u3064\u3051\u308b\u305f\u3081\u306b\u30b0\u30e9\u30d5\u3092\u63cf\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n
\u4ee5\u4e0b\u3067\u306f\u4f8b\u3068\u3057\u3066\u30d9\u30c3\u30bb\u30eb\u95a2\u6570 $J_0(x) = $ besselj(0,x)<\/code> \u306e\u30bc\u30ed\u70b9\u3092\u6c42\u3081\u3066\u307f\u307e\u3059\u3002\u30d9\u30c3\u30bb\u30eb\u95a2\u6570 besselj()<\/code> \u306f<\/p>\n
\n
from<\/span> sympy<\/span> import<\/span> *<\/span>\r\n<\/pre>\n<\/div>\n
\u3059\u308c\u3070\u4f7f\u3048\u307e\u3059\u3002$x$ \u8ef8\u3092\u6a2a\u5207\u308b\u70b9\u304c\u308f\u304b\u308a\u3084\u3059\u3044\u3088\u3046\u306b\uff0c$y=0$ \uff08$x$ \u8ef8\uff09\u3082\u63cf\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[13]:<\/div>\n
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graphics<\/span>(<\/span>\r\n    line<\/span>(<\/span>besselj<\/span>(<\/span>0<\/span>,<\/span> x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> 0<\/span>,<\/span> 10<\/span>)),<\/span> \r\n    line<\/span>(<\/span>0<\/span>,<\/span> (<\/span>x<\/span>,<\/span> 0<\/span>,<\/span> 10<\/span>),<\/span> ''<\/span>),<\/span> # x \u8ef8<\/span>\r\n    xlabel<\/span> =<\/span> \"$x$\"<\/span>,<\/span> ylabel<\/span> =<\/span> \"$J_0(x)$\"<\/span>,<\/span> \r\n    xlim<\/span> =<\/span> [<\/span>0<\/span>,<\/span> 10<\/span>],<\/span> \r\n    title<\/span> =<\/span> \"\u30d9\u30c3\u30bb\u30eb\u95a2\u6570 $J_0(x)$\"<\/span>\r\n)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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<\/p>\n
\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u3092\u6c42\u3081\u3066\u304f\u308c\u308b\u95a2\u6570\u306f nsolve()<\/code> \u3067\u3059\u3002$x=2$ \u306e\u8fd1\u304f\u3067 $x$ \u8ef8\u3068\u4ea4\u308f\u3063\u3066\u3044\u307e\u3059\u304b\u3089\uff0c\u3053\u306e\u3042\u305f\u308a\u306e\u89e3\u3092\u63a2\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[14]:<\/div>\n
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# besselj(0,x) = 0 \u3068\u306a\u308b\u89e3\u3092<\/span>\r\n# x \u304c 2 \u306e\u3042\u305f\u308a\u3067\u63a2\u3059<\/span>\r\n\r\nnsolve<\/span>(<\/span>besselj<\/span>(<\/span>0<\/span>,<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 2<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[14]:<\/div>\n
$\\displaystyle 2.40482555769577$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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[2, 4]<\/code> \u306e\u3088\u3046\u306b\u7bc4\u56f2\u3092\u6307\u5b9a\u3057\u3066\u89e3\u3092\u63a2\u3059\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[15]:<\/div>\n
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# 2 <= x <= 4 \u306e\u9593\u3067<\/span>\r\n# besselj(0, x) = 0 \u3068\u306a\u308b x \u3092\u63a2\u3059<\/span>\r\n\r\nnsolve<\/span>(<\/span>besselj<\/span>(<\/span>0<\/span>,<\/span>x<\/span>),<\/span> x<\/span>,<\/span> [<\/span>2<\/span>,<\/span> 4<\/span>])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[15]:<\/div>\n
$\\displaystyle 2.40482555769577$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[16]:<\/div>\n
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# \u307b\u3093\u3068\u306b\u89e3\u306b\u306a\u3063\u3066\u308b\u304b\u78ba\u8a8d\u3002\u30bc\u30ed\u306b\u8fd1\u3051\u308c\u3070\u3088\u3044\u3002<\/span>\r\nbesselj<\/span>(<\/span>0<\/span>,<\/span>_<\/span>)<\/span>.<\/span>evalf<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
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Out[16]:<\/div>\n
$\\displaystyle -6.10876525973673 \\cdot 10^{-17}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
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\u25cb\u7df4\u7fd2\uff1a\u95a2\u6570\u306e\u6975\u5927\u5024<\/h4>\n
    \n
  1. $\\displaystyle f(x) = \\frac{x^3}{e^x -1}$ \u304c\u6975\u5927\u5024\u3092\u3068\u308b\u3068\u304d\u306e $x$ \u306e\u5024\uff0c\u304a\u3088\u3073\u305d\u306e\u3068\u304d\u306e\u6975\u5927\u5024\u3092\u6c42\u3081\u308b\u3002<\/li>\n
  2. $\\displaystyle g(x) = \\frac{x^5}{e^x -1}$ \u304c\u6975\u5927\u5024\u3092\u3068\u308b\u3068\u304d\u306e $x$ \u306e\u5024\uff0c\u304a\u3088\u3073\u305d\u306e\u3068\u304d\u306e\u6975\u5927\u5024\u3082\u540c\u69d8\u306b\u3057\u3066\u6c42\u3081\u3066\u307f\u3088\u3002<\/li>\n<\/ol>\n
    \u307e\u305a $f(x)$ \u306e\u6570\u5024\u5fae\u5206 ndiff(f, x)<\/code> \u3092\u30b0\u30e9\u30d5\u306b\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
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    In\u00a0[17]:<\/div>\n
    \n
    \n
    \n
    def<\/span> f<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> x<\/span>**<\/span>3<\/span>\/<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>)<\/span> -<\/span> 1<\/span>)<\/span>\r\n\r\ngraphics<\/span>(<\/span>\r\n    line<\/span>(<\/span>ndiff<\/span>(<\/span>f<\/span>,<\/span> x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> 0<\/span>,<\/span> 5<\/span>),<\/span> '$f(x)$ \u306e\u6570\u5024\u5fae\u5206'<\/span>),<\/span> \r\n    line<\/span>(<\/span>0<\/span>,<\/span> (<\/span>x<\/span>,<\/span> 0<\/span>,<\/span> 5<\/span>),<\/span> ''<\/span>),<\/span> \r\n    xlabel<\/span>=<\/span>'$x$'<\/span>,<\/span> ylabel<\/span>=<\/span>'$y$'<\/span>\r\n)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    <\/div>\n
    \n
    <\/p>\n
    \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    ndiff(f, x)<\/code> \u306f $2 < x <3$ \u306e\u3042\u305f\u308a\u3067\u30bc\u30ed\u306b\u306a\u3063\u3066\u3044\u308b\u306e\u3067\uff0c\u305d\u306e\u3042\u305f\u308a\u3092 nsolve()<\/code> \u3067\u63a2\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[18]:<\/div>\n
    \n
    \n
    \n
    nsolve<\/span>(<\/span>ndiff<\/span>(<\/span>f<\/span>,<\/span> x<\/span>),<\/span> x<\/span>,<\/span> [<\/span>2<\/span>,<\/span> 3<\/span>])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[18]:<\/div>\n
    $\\displaystyle 2.82143937214806$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[19]:<\/div>\n
    \n
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    \n
    # \u305d\u306e\u3068\u304d\u306e f(x) \u306e\u6975\u5927\u5024\u306f...<\/span>\r\nf<\/span>(<\/span>_<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[19]:<\/div>\n
    $\\displaystyle 1.42143547274774$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u5ff5\u306e\u305f\u3081\uff0c\u6570\u5024\u5fae\u5206 ndiff(f, x)<\/code> \u3067\u306f\u306a\u304f\uff0c\u89e3\u6790\u7684\u306a\u5fae\u5206 df<\/code> $\\displaystyle = \\frac{df}{dx}$ \u304c\u30bc\u30ed\u3068\u306a\u308b\u70b9\u3082 nsolve()<\/code> \u3067\u63a2\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[20]:<\/div>\n
    \n
    \n
    \n
    def<\/span> f<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> x<\/span>**<\/span>3<\/span>\/<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>)<\/span> -<\/span> 1<\/span>)<\/span>\r\n\r\ndf<\/span> =<\/span> diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>\r\ndisplay<\/span>(<\/span>df<\/span>)<\/span>\r\n\r\nnsolve<\/span>(<\/span>df<\/span>,<\/span> x<\/span>,<\/span> [<\/span>2<\/span>,<\/span> 3<\/span>])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    <\/div>\n
    $\\displaystyle – \\frac{x^{3} e^{x}}{\\left(e^{x} -1\\right)^{2}} + \\frac{3 x^{2}}{e^{x} -1}$<\/div>\n<\/div>\n
    \n
    Out[20]:<\/div>\n
    $\\displaystyle 2.82143937212208$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[21]:<\/div>\n
    \n
    \n
    \n
    # \u305d\u306e\u3068\u304d\u306e f(x) \u306e\u6975\u5927\u5024\u306f...<\/span>\r\nf<\/span>(<\/span>_<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[21]:<\/div>\n
    $\\displaystyle 1.42143547274774$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306b\u3088\u308b1\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u6cd5<\/h3>\n
    \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u6cd5\u3092\u3057\u3066\u304f\u308c\u308b\u95a2\u6570\u306f SymPy \u306b\u306f\u7528\u610f\u3055\u308c\u3066\u3044\u306a\u3044\u3088\u3046\u3067\u3059\u3002\u5099\u3048\u4ed8\u3051\u306e\u95a2\u6570\u306b\u983c\u3089\u305a\uff0c1\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/p>\n
    $$ \\frac{dx}{dt} = f(t, x)$$<\/p>\n
    \u30924\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u89e3\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n
    \u521d\u671f\u6761\u4ef6\u3092 $t=t_0$ \u3067 $x(t_0) = x_0$ \u3068\u3057\uff0c$t = t_1$ \u307e\u3067\u3092 $N$ \u5206\u5272\u3057\u3066\u89e3\u304d\u307e\u3059\u3002<\/p>\n
    $$ h = \\frac{t_1 -t_0}{N}, \\quad t_i = t_0 + i\\, h,
    \n\\quad x_i = x(t_i)$$<\/p>\n
    \u3068\u3057\uff0c\u4ee5\u4e0b\u306e\u5f0f\u304b\u3089\u6b21\u306e\u30b9\u30c6\u30c3\u30d7\u306e\u5024 $x_{i+1}$ \u3092\u6c7a\u3081\u307e\u3059\u3002<\/p>\n
    \\begin{eqnarray}
    \nk_1 &=& h\\, f(t_i, x_i) \\\\
    \nk_2 &=& h\\, f(t_i + h\/2, x_i + k_1\/2) \\\\
    \nk_3 &=& h\\, f(t_i + h\/2, x_i + k_2\/2) \\\\
    \nk_4 &=& h\\, f(t_i + h, x_i + k_3) \\\\
    \nx_{i+1} &=& x_i + \\frac{1}{6} (k_1 + 2 k_2 + 2 k_3 + k_4)
    \n\\end{eqnarray}<\/p>\n
    \u4ee5\u4e0a\u306e\u3088\u3046\u306a\u8a08\u7b97\u3092\u884c\u3046\u95a2\u6570 rk1()<\/code> \u3092 Python \u306e\u95a2\u6570\u3068\u3057\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[22]:<\/div>\n
    \n
    \n
    \n
    def<\/span> rk1<\/span>(<\/span>func<\/span>,<\/span> x0<\/span>,<\/span> t0<\/span>,<\/span> t1<\/span>,<\/span> N<\/span>):<\/span>\r\n    \"\"\" dx\/dt = func(t, x) \u3092<\/span>\r\n        \u521d\u671f\u6761\u4ef6 t = t0 \u3067 x = x0 \u3068\u3057\uff0c<\/span>\r\n        [t0, t1] \u3092 N \u5206\u5272\u3057\u3066<\/span>\r\n        4\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u89e3\u304f<\/span>\r\n    \"\"\"<\/span>\r\n    h<\/span> =<\/span> (<\/span>t1<\/span> -<\/span> t0<\/span>)<\/span>\/<\/span>N<\/span>\r\n    T<\/span> =<\/span> [<\/span>t0<\/span>]<\/span>\r\n    X<\/span> =<\/span> [<\/span>x0<\/span>]<\/span>\r\n    t<\/span> =<\/span> t0<\/span>\r\n    x<\/span> =<\/span> x0<\/span>\r\n    for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>N<\/span>):<\/span>\r\n        k1<\/span> =<\/span> h<\/span>*<\/span>func<\/span>(<\/span>t<\/span>,<\/span> x<\/span>)<\/span>\r\n        k2<\/span> =<\/span> h<\/span>*<\/span>func<\/span>(<\/span>t<\/span> +<\/span> h<\/span>\/<\/span>2<\/span>,<\/span> x<\/span> +<\/span> k1<\/span>\/<\/span>2<\/span>)<\/span>\r\n        k3<\/span> =<\/span> h<\/span>*<\/span>func<\/span>(<\/span>t<\/span> +<\/span> h<\/span>\/<\/span>2<\/span>,<\/span> x<\/span> +<\/span> k2<\/span>\/<\/span>2<\/span>)<\/span>\r\n        k4<\/span> =<\/span> h<\/span>*<\/span>func<\/span>(<\/span>t<\/span> +<\/span> h<\/span>,<\/span> x<\/span> +<\/span> k3<\/span>)<\/span>\r\n        x<\/span> =<\/span> x<\/span> +<\/span> (<\/span>k1<\/span> +<\/span> 2<\/span>*<\/span>k2<\/span> +<\/span> 2<\/span>*<\/span>k3<\/span> +<\/span> k4<\/span>)<\/span>\/<\/span>6<\/span>\r\n        t<\/span> =<\/span> t<\/span> +<\/span> h<\/span>\r\n        T<\/span>.<\/span>append<\/span>(<\/span>t<\/span>)<\/span>\r\n        X<\/span>.<\/span>append<\/span>(<\/span>x<\/span>)<\/span>\r\n    return<\/span> [<\/span>T<\/span>,<\/span> X<\/span>]<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u4f8b\u3068\u3057\u3066\uff0c\u7c21\u5358\u306a\u30ed\u30b8\u30b9\u30c6\u30a3\u30c3\u30af\u65b9\u7a0b\u5f0f<\/p>\n
    $$ \\frac{dx}{dt} = f(t, x) = (1-x)\\,x$$<\/p>\n
    \u3092\uff0c\u521d\u671f\u6761\u4ef6 $t_0 = 0$ \u3067 $x_0 = x(t_0) = 0.1$\uff0c$t = t_0 = 0$ \u304b\u3089 $t_1 = 10$ \u307e\u3067\u3092 $ N$ \u5206\u5272\u3057\u3066\u89e3\u304d\u307e\u3059\u3002<\/p>\n
    \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u7cbe\u5ea6\u306f\u5206\u5272\u6570 $N$ \u306b\u4f9d\u5b58\u3057\u307e\u3059\u3002$N$ \u306e\u5024\u3092\u5909\u3048\u3066 $x(t_1)$ \u306e\u5024\u3092\u8a08\u7b97\u3057\u3066\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[23]:<\/div>\n
    \n
    \n
    \n
    # dx\/dt = f(t, x) \u3092\u89e3\u304f\u3002<\/span>\r\ndef<\/span> f<\/span>(<\/span>t<\/span>,<\/span> x<\/span>):<\/span>\r\n    return<\/span> (<\/span>1<\/span>-<\/span>x<\/span>)<\/span>*<\/span>x<\/span>\r\n\r\nx0<\/span> =<\/span> 0.1<\/span>\r\nt0<\/span> =<\/span> 0<\/span>\r\nt1<\/span> =<\/span> 10<\/span>\r\n\r\nfor<\/span> N<\/span> in<\/span> [<\/span>100<\/span>,<\/span> 200<\/span>,<\/span> 400<\/span>,<\/span> 800<\/span>,<\/span> 1600<\/span>]:<\/span>\r\n    T1<\/span>,<\/span> X1<\/span> =<\/span> rk1<\/span>(<\/span>f<\/span>,<\/span> x0<\/span>,<\/span> t0<\/span>,<\/span> t1<\/span>,<\/span> N<\/span>)<\/span>\r\n    print<\/span>(<\/span>\"N =<\/span>%5d<\/span>, x(t1) =<\/span>%15.12f<\/span>\"<\/span> %<\/span> (<\/span>N<\/span>,<\/span> X1<\/span>[<\/span>-<\/span>1<\/span>]))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    <\/div>\n
    \n
    N =  100, x(t1) = 0.999591565222\r\nN =  200, x(t1) = 0.999591567379\r\nN =  400, x(t1) = 0.999591567509\r\nN =  800, x(t1) = 0.999591567517\r\nN = 1600, x(t1) = 0.999591567517\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u4e0a\u306e\u7d50\u679c\u304b\u3089\uff0c$N=800$ \u304f\u3089\u3044\u3067\u5c0f\u6570\u70b9\u4ee5\u4e0b $12$ \u6841\u7a0b\u5ea6\u306e\u7cbe\u5ea6\u304c\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u3067\u306f\u3053\u306e\u304f\u3089\u3044\u306e $N$ \u3067\u8a08\u7b97\u3092\u3059\u308b\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[24]:<\/div>\n
    \n
    \n
    \n
    # dx\/dt = f(t, x) \u3092\u89e3\u304f\u3002<\/span>\r\ndef<\/span> f<\/span>(<\/span>t<\/span>,<\/span> x<\/span>):<\/span>\r\n    return<\/span> (<\/span>1<\/span>-<\/span>x<\/span>)<\/span>*<\/span>x<\/span>\r\n\r\nx0<\/span> =<\/span> 0.1<\/span>\r\nt0<\/span> =<\/span> 0<\/span>\r\nt1<\/span> =<\/span> 10<\/span>\r\n\r\nndiv<\/span> =<\/span> 40<\/span>\r\nNdiv<\/span> =<\/span> 20<\/span>\r\nN<\/span> =<\/span> ndiv<\/span> *<\/span> Ndiv<\/span>\r\n\r\nT1<\/span>,<\/span> X1<\/span> =<\/span> rk1<\/span>(<\/span>f<\/span>,<\/span> x0<\/span>,<\/span> t0<\/span>,<\/span> t1<\/span>,<\/span> N<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    T1<\/code>\uff0cX1<\/code> \u306b\u306f N+1<\/code> \u70b9\u306e\u6570\u5024\u89e3\u306e\u5024\u304c\u4ee3\u5165\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u6ed1\u3089\u304b\u306a\u66f2\u7dda\u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u306b\u306f\uff0c\u5168\u90e8\u306e\u70b9\u3092\u4f7f\u3048\u3070\u3044\u3044\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[25]:<\/div>\n
    \n
    \n
    \n
    graphics<\/span>(<\/span>\r\n    list_2d<\/span>(<\/span>T1<\/span>,<\/span> X1<\/span>),<\/span> \r\n    xlabel<\/span> =<\/span> \"$t$\"<\/span>,<\/span> ylabel<\/span> =<\/span> \"$x(t)$\"<\/span>,<\/span>\r\n    title<\/span> =<\/span> \"\u30ed\u30b8\u30b9\u30c6\u30a3\u30c3\u30af\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\"<\/span>\r\n)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    <\/div>\n
    \n
    <\/p>\n
    \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u3053\u3053\u3067\u306f\u5c11\u3057\u9593\u5f15\u3044\u3066 Ndiv+1<\/code> \u500b\u306e\u70b9\u3092\u30b0\u30e9\u30d5\u306b\u3057\u3066\u307f\u307e\u3059\u3002\uff08\u3042\u3068\u3067\u89e3\u6790\u89e3\u3068\u6bd4\u3079\u3066\u307f\u308b\u305f\u3081\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[26]:<\/div>\n
    \n
    \n
    \n
    # \u30c7\u30fc\u30bf\u3092\u9593\u5f15\u304f<\/span>\r\nt1<\/span> =<\/span> [<\/span>T1<\/span>[<\/span>i<\/span>]<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>0<\/span>,<\/span> len<\/span>(<\/span>T1<\/span>),<\/span> ndiv<\/span>)]<\/span>\r\nx1<\/span> =<\/span> [<\/span>X1<\/span>[<\/span>i<\/span>]<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>0<\/span>,<\/span> len<\/span>(<\/span>X1<\/span>),<\/span> ndiv<\/span>)]<\/span>\r\n\r\nprint<\/span>(<\/span>'\u5143\u306e\u8981\u7d20\u6570\u306f'<\/span>,<\/span>len<\/span>(<\/span>X1<\/span>))<\/span>\r\nprint<\/span>(<\/span>'\u9593\u5f15\u304f\u3068\u3000\u3000'<\/span>,<\/span>len<\/span>(<\/span>x1<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    <\/div>\n
    \n
    \u5143\u306e\u8981\u7d20\u6570\u306f 801\r\n\u9593\u5f15\u304f\u3068\u3000\u3000 21\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[27]:<\/div>\n
    \n
    \n
    \n
    # \u30c7\u30fc\u30bf\u3092\u9593\u5f15\u3044\u3066\u30b0\u30e9\u30d5\u306b<\/span>\r\n\r\ngraphics<\/span>(<\/span>\r\n    list_2d<\/span>(<\/span>t1<\/span>,<\/span> x1<\/span>,<\/span> scatter<\/span> =<\/span> True<\/span>),<\/span> \r\n    xlabel<\/span> =<\/span> \"$t$\"<\/span>,<\/span> ylabel<\/span> =<\/span> \"$x(t)$\"<\/span>,<\/span>\r\n    title<\/span> =<\/span> \"\u30ed\u30b8\u30b9\u30c6\u30a3\u30c3\u30af\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\"<\/span>\r\n)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    <\/div>\n
    \n
    <\/p>\n
    \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u53c2\u8003\uff1aSymPy \u306b\u3088\u308b1\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u89e3\u6790\u89e3<\/h4>\n
    \u7c21\u5358\u306a\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5834\u5408\u306f SymPy \u306e dsolve()<\/code> \u3067\u89e3\u6790\u7684\u306b\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u4f8b\u3068\u3057\u3066\uff0c\u7c21\u5358\u306a\u30ed\u30b8\u30b9\u30c6\u30a3\u30c3\u30af\u65b9\u7a0b\u5f0f<\/p>\n
    $$ \\frac{dx}{dt} = (1-x)\\,x$$<\/p>\n
    \u3092\uff0c\u521d\u671f\u6761\u4ef6 $t = 0$ \u3067 $x(0) = 0.1$ \u3068\u3057\u3066\u89e3\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[28]:<\/div>\n
    \n
    \n
    \n
    # \u89e3\u304f\u3079\u304d\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092 eq \u3068\u3057\u3066\u5b9a\u7fa9<\/span>\r\nvar<\/span>(<\/span>'t'<\/span>)<\/span>\r\nx<\/span> =<\/span> Function<\/span>(<\/span>'x'<\/span>)(<\/span>t<\/span>)<\/span>\r\neq<\/span> =<\/span> Eq<\/span>(<\/span>Derivative<\/span>(<\/span>x<\/span>,<\/span> t<\/span>),<\/span> (<\/span>1<\/span>-<\/span>x<\/span>)<\/span>*<\/span>x<\/span>)<\/span>\r\neq<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[28]:<\/div>\n
    $\\displaystyle \\frac{d}{d t} x{\\left(t \\right)} = \\left(1 -x{\\left(t \\right)}\\right) x{\\left(t \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[29]:<\/div>\n
    \n
    \n
    \n
    # \u521d\u671f\u6761\u4ef6\u3092\u7279\u306b\u8a2d\u5b9a\u305b\u305a\u306b\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f<\/span>\r\ndsolve<\/span>(<\/span>eq<\/span>,<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[29]:<\/div>\n
    $\\displaystyle x{\\left(t \\right)} = \\frac{1}{C_{1} e^{-t} + 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[30]:<\/div>\n
    \n
    \n
    \n
    # \u521d\u671f\u6761\u4ef6\u3092 t=0 \u3067 x(0) = 1\/10 \u3068\u3057\u3066\u89e3\u304f<\/span>\r\nsol1<\/span> =<\/span> dsolve<\/span>(<\/span>eq<\/span>,<\/span> x<\/span>,<\/span> ics<\/span>=<\/span>{<\/span>x<\/span>.<\/span>subs<\/span>(<\/span>t<\/span>,<\/span>0<\/span>):<\/span> Rational<\/span>(<\/span>1<\/span>,<\/span>10<\/span>)})<\/span>\r\nsol1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[30]:<\/div>\n
    $\\displaystyle x{\\left(t \\right)} = \\frac{1}{1 + 9 e^{-t}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[31]:<\/div>\n
    \n
    \n
    \n
    # ans \u306e\u53f3\u8fba\u3092\u8868\u793a <\/span>\r\n# rhs \u306f rirhgt-hand-side<\/span>\r\nsimplify<\/span>(<\/span>sol1<\/span>.<\/span>rhs<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[31]:<\/div>\n
    $\\displaystyle \\frac{e^{t}}{e^{t} + 9}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    … \u3068\u3044\u3046\u3053\u3068\u3067\uff0c\u89e3\u6790\u89e3\u306f<\/p>\n
    $$x(t) = \\frac{\\exp(t)}{9 + \\exp(t)}$$<\/p>\n
    \u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002
    \n\u5f97\u3089\u308c\u305f\u89e3\u6790\u89e3 $x(t)$ \u3068\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u5f97\u3089\u308c\u305f\u6570\u5024\u89e3\u3092\u30b0\u30e9\u30d5\u306b\u3059\u308b\u4f8b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[32]:<\/div>\n
    \n
    \n
    \n
    graphics<\/span>(<\/span>\r\n    line<\/span>(<\/span>sol1<\/span>.<\/span>rhs<\/span>,<\/span> (<\/span>t<\/span>,<\/span> 0<\/span>,<\/span> 10<\/span>),<\/span> '\u89e3\u6790\u89e3'<\/span>),<\/span> \r\n    list_2d<\/span>(<\/span>t1<\/span>,<\/span> x1<\/span>,<\/span> '\u6570\u5024\u89e3'<\/span>,<\/span> scatter<\/span> =<\/span> True<\/span>,<\/span> \r\n            # \"ms\" \u3064\u307e\u308a \"markersize\" \u3092\u5c0f\u3055\u3081\u306b<\/span>\r\n            rendering_kw<\/span>=<\/span>{<\/span>\"ms\"<\/span>:<\/span>3<\/span>}),<\/span> \r\n    xlabel<\/span> =<\/span> \"$t$\"<\/span>,<\/span> ylabel<\/span> =<\/span> \"$x(t)$\"<\/span>,<\/span>\r\n    title<\/span> =<\/span> \"\u30ed\u30b8\u30b9\u30c6\u30a3\u30c3\u30af\u65b9\u7a0b\u5f0f\u306e\u89e3\"<\/span>\r\n)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    <\/div>\n
    \n
    <\/p>\n
    \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u5f97\u3089\u308c\u305f $x(t)$ \u3092\u3069\u3046\u3084\u3063\u3066\u95a2\u6570\u306b\u5b9a\u7fa9\u3059\u308b\u304b\u60a9\u307f\u307e\u3059\u3002\u4ee5\u4e0b\u306e\u4f8b\u3067\u306a\u3093\u3068\u304b\u306a\u308a\u307e\u3059\u304c\uff0c\u5c11\u3057\u30c8\u30ea\u30c3\u30ad\u30fc\u3067\u3059\u304b\u306d\u3047\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[33]:<\/div>\n
    \n
    \n
    \n
    def<\/span> xsol1<\/span>(<\/span>s<\/span>):<\/span>\r\n    return<\/span> sol1<\/span>.<\/span>rhs<\/span>.<\/span>subs<\/span>(<\/span>t<\/span>,<\/span>s<\/span>)<\/span>\r\n\r\nxsol1<\/span>(<\/span>t<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[33]:<\/div>\n
    $\\displaystyle \\frac{1}{1 + 9 e^{-t}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306b\u3088\u308b2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u6cd5<\/h3>\n
    \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u6cd5\u3092\u3057\u3066\u304f\u308c\u308b\u95a2\u6570\u306f SymPy \u306b\u306f\u7528\u610f\u3055\u308c\u3066\u3044\u306a\u3044\u3088\u3046\u3067\u3059\u3002\u5099\u3048\u4ed8\u3051\u306e\u95a2\u6570\u306b\u983c\u3089\u305a\uff0c\u6b21\u306e\u3088\u3046\u306a2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u89e3\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n
    $$ \\frac{d^2 x}{dt^2 } = F\\left(t, x, \\frac{dx}{dt}\\right)$$<\/p>\n
    \u3053\u306e\u5834\u5408\u306b\u306f\uff0c $\\displaystyle v \\equiv \\frac{dx}{dt}$ \u3068\u304a\u3051\u3070\uff0c\u6b21\u306e\u3088\u3046\u306a\u9023\u7acb1\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5f62\u306b\u5e30\u7740\u3067\u304d\u308b\u3002<\/p>\n
    \\begin{eqnarray}
    \n\\frac{dx}{dt} &=& F_1(t, x, v) = v \\\\
    \n\\frac{dv}{dt} &=& F_2(t, x, v) = F(t, x, v)
    \n\\end{eqnarray}<\/p>\n
    \u521d\u671f\u6761\u4ef6\u3092 $t=t_0$ \u3067 $x(t_0) = x_0, v(t_0) = v_0$ \u3068\u3057\uff0c$t = t_1$ \u307e\u3067\u3092 $N$ \u5206\u5272\u3057\u3066\u89e3\u304d\u307e\u3059\u3002<\/p>\n
    $$ h = \\frac{t_1 -t_0}{N}, \\quad t_i = t_0 + i\\, h,
    \n\\quad x_i = x(t_i), \\quad v_i = v(t_i)$$<\/p>\n
    \u3068\u3057\uff0c\u4ee5\u4e0b\u306e\u5f0f\u304b\u3089\u6b21\u306e\u30b9\u30c6\u30c3\u30d7\u306e\u5024 $x_{i+1}, v_{i+1}$ \u3092\u6c7a\u3081\u307e\u3059\u3002<\/p>\n
    \\begin{eqnarray}
    \nk_1 &=& h\\, F_1(t_i, x_i, v_i) = h\\,v_i \\\\
    \nm_1 &=& h\\, F_2(t_i, x_i, v_i) \\\\
    \nk_2 &=& h\\, F_1(t_i + h\/2, x_i + k_1\/2, v_i + m_1\/2) = h\\,(v_i + m_1\/2)\\\\
    \nm_2 &=& h\\, F_2(t_i + h\/2, x_i + k_1\/2, v_i + m_1\/2) \\\\
    \nk_3 &=& h\\, F_1(t_i + h\/2, x_i + k_2\/2, v_i + m_2\/2) = h\\,(v_i + m_2\/2)\\\\
    \nm_3 &=& h\\, F_2(t_i + h\/2, x_i + k_2\/2, v_i + m_2\/2) \\\\
    \nk_4 &=& h\\, F_1(t_i + h, x_i + k_3, v_i + m_3) = h\\,(v_i + m_3)\\\\
    \nm_4 &=& h\\, F_2(t_i + h, x_i + k_3, v_i + m_3) \\\\
    \nx_{i+1} &=& x_i + \\frac{1}{6} (k_1 + 2 k_2 + 2 k_3 + k_4)\\\\
    \nv_{i+1} &=& v_i + \\frac{1}{6} (m_1 + 2 m_2 + 2 m_3 + m_4)
    \n\\end{eqnarray}<\/p>\n
    \u4ee5\u4e0a\u306e\u3088\u3046\u306a\u8a08\u7b97\u3092\u884c\u3046\u95a2\u6570 rk2()<\/code> \u3092 Python \u306e\u95a2\u6570\u3068\u3057\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[34]:<\/div>\n
    \n
    \n
    \n
    def<\/span> rk2<\/span>(<\/span>Func<\/span>,<\/span> x0<\/span>,<\/span> v0<\/span>,<\/span> t0<\/span>,<\/span> t1<\/span>,<\/span> N<\/span>):<\/span>\r\n    \"\"\" d^2 x\/dt^2 = Func(t, x, v) \u3092<\/span>\r\n        \u521d\u671f\u6761\u4ef6 t = t0 \u3067 x = x0, v = v0 \u3068\u3057\uff0c<\/span>\r\n        [t0, t1] \u3092 N \u5206\u5272\u3057\u3066<\/span>\r\n        4\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u89e3\u304f<\/span>\r\n    \"\"\"<\/span>\r\n    h<\/span> =<\/span> float<\/span>((<\/span>t1<\/span> -<\/span> t0<\/span>)<\/span>\/<\/span>N<\/span>)<\/span>\r\n    T<\/span> =<\/span> [<\/span>t0<\/span>]<\/span>\r\n    X<\/span> =<\/span> [<\/span>x0<\/span>]<\/span>\r\n    V<\/span> =<\/span> [<\/span>v0<\/span>]<\/span>\r\n    t<\/span> =<\/span> t0<\/span>\r\n    x<\/span> =<\/span> x0<\/span>\r\n    v<\/span> =<\/span> v0<\/span>\r\n    for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>N<\/span>):<\/span>\r\n        k1<\/span> =<\/span> h<\/span>*<\/span>v<\/span>\r\n        m1<\/span> =<\/span> h<\/span>*<\/span>Func<\/span>(<\/span>t<\/span>,<\/span> x<\/span>,<\/span> v<\/span>)<\/span>\r\n        k2<\/span> =<\/span> h<\/span>*<\/span>(<\/span>v<\/span>+<\/span>m1<\/span>\/<\/span>2<\/span>)<\/span>\r\n        m2<\/span> =<\/span> h<\/span>*<\/span>Func<\/span>(<\/span>t<\/span>+<\/span>h<\/span>\/<\/span>2<\/span>,<\/span> x<\/span>+<\/span>k1<\/span>\/<\/span>2<\/span>,<\/span> v<\/span>+<\/span>m1<\/span>\/<\/span>2<\/span>)<\/span>\r\n        k3<\/span> =<\/span> h<\/span>*<\/span>(<\/span>v<\/span>+<\/span>m2<\/span>\/<\/span>2<\/span>)<\/span>\r\n        m3<\/span> =<\/span> h<\/span>*<\/span>Func<\/span>(<\/span>t<\/span>+<\/span>h<\/span>\/<\/span>2<\/span>,<\/span> x<\/span>+<\/span>k2<\/span>\/<\/span>2<\/span>,<\/span> v<\/span>+<\/span>m2<\/span>\/<\/span>2<\/span>)<\/span>\r\n        k4<\/span> =<\/span> h<\/span>*<\/span>(<\/span>v<\/span>+<\/span>m3<\/span>)<\/span>\r\n        m4<\/span> =<\/span> h<\/span>*<\/span>Func<\/span>(<\/span>t<\/span>+<\/span>h<\/span>,<\/span> x<\/span>+<\/span>k3<\/span>,<\/span> v<\/span>+<\/span>m3<\/span>)<\/span>\r\n        x<\/span> =<\/span> x<\/span> +<\/span> (<\/span>k1<\/span> +<\/span> 2<\/span>*<\/span>k2<\/span> +<\/span> 2<\/span>*<\/span>k3<\/span> +<\/span> k4<\/span>)<\/span>\/<\/span>6<\/span>\r\n        v<\/span> =<\/span> v<\/span> +<\/span> (<\/span>m1<\/span> +<\/span> 2<\/span>*<\/span>m2<\/span> +<\/span> 2<\/span>*<\/span>m3<\/span> +<\/span> m4<\/span>)<\/span>\/<\/span>6<\/span>\r\n        t<\/span> =<\/span> t<\/span> +<\/span> h<\/span> \r\n        T<\/span>.<\/span>append<\/span>(<\/span>t<\/span>)<\/span>\r\n        X<\/span>.<\/span>append<\/span>(<\/span>x<\/span>)<\/span>\r\n        V<\/span>.<\/span>append<\/span>(<\/span>v<\/span>)<\/span>\r\n    return<\/span> [<\/span>T<\/span>,<\/span> X<\/span>,<\/span> V<\/span>]<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u4f8b\u3068\u3057\u3066\uff0c\u7c21\u5358\u306a\u6e1b\u8870\uff0b\u5f37\u5236\u632f\u52d5\u306e\u65b9\u7a0b\u5f0f<\/p>\n
    $$ \\frac{d^2 x}{dt^2} = F(t, x, v) = -x -a v + b \\cos(t), \\quad v = \\frac{dx}{dt} $$<\/p>\n
    \u3092\uff0c<\/p>\n
      \n
    • $a , b $ \u306b\u3044\u304f\u3064\u304b\u5024\u3092\u5165\u308c\u3066<\/li>\n
    • \u521d\u671f\u6761\u4ef6 $t_0 = 0$ \u3067 $x_0 = x(t_0) = 3, v_0 = v(t_0) = 0$\uff0c<\/li>\n
    • $t = t_0 = 0$ \u304b\u3089 $t_1 = 20$ \u307e\u3067\u3092 $ N $ \u5206\u5272<\/li>\n<\/ul>\n
      \u3057\u3066\u89e3\u304d\u307e\u3059\u3002<\/p>\n
      $N$ \u306e\u5024\u3092\u3044\u304f\u3064\u304b\u5909\u3048\u3066\u307f\u3066\uff0c\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002\uff08\u7d50\u69cb\u6642\u9593\u304c\u304b\u304b\u308a\u307e\u3059\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[35]:<\/div>\n
      \n
      \n
      \n
      def<\/span> F<\/span>(<\/span>t<\/span>,<\/span> x<\/span>,<\/span> v<\/span>):<\/span>\r\n    global<\/span> a<\/span>,<\/span> b<\/span>\r\n    return<\/span> -<\/span>x<\/span> -<\/span> a<\/span>*<\/span>v<\/span> +<\/span> b<\/span>*<\/span>cos<\/span>(<\/span>t<\/span>)<\/span>\r\n\r\na<\/span> =<\/span> 0.5<\/span>\r\nb<\/span> =<\/span> 0.2<\/span>\r\nx0<\/span> =<\/span> 3<\/span>\r\nv0<\/span> =<\/span> 0<\/span>\r\nt0<\/span> =<\/span> 0<\/span>\r\nt1<\/span> =<\/span> 20<\/span>\r\n\r\nfor<\/span> N<\/span> in<\/span> [<\/span>200<\/span>,<\/span> 400<\/span>,<\/span> 800<\/span>,<\/span> 1600<\/span>]:<\/span>\r\n    T2<\/span>,<\/span> X2<\/span>,<\/span> V2<\/span> =<\/span> rk2<\/span>(<\/span>F<\/span>,<\/span> x0<\/span>,<\/span> v0<\/span>,<\/span> t0<\/span>,<\/span> t1<\/span>,<\/span> N<\/span>)<\/span>\r\n    # X2 \u306e\u6700\u5f8c\u306e\u8981\u7d20 X2[-1] \u3092\u8868\u793a<\/span>\r\n    print<\/span>(<\/span>\"N =<\/span>%5d<\/span>, x(t1) =<\/span>%15.12f<\/span>\"<\/span> %<\/span> (<\/span>N<\/span>,<\/span> X2<\/span>[<\/span>-<\/span>1<\/span>]))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      <\/div>\n
      \n
      N =  200, x(t1) = 0.383967288338\r\nN =  400, x(t1) = 0.383966889383\r\nN =  800, x(t1) = 0.383966861347\r\nN = 1600, x(t1) = 0.383966859500\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u4e0a\u306e\u7d50\u679c\u304b\u3089\uff0c$N=800$ \u304f\u3089\u3044\u3067\u5c0f\u6570\u70b9\u4ee5\u4e0b $8$ \u6841\u7a0b\u5ea6\u306e\u7cbe\u5ea6\u304c\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u3067\u306f\u3053\u306e\u304f\u3089\u3044\u306e $N$ \u3067\u8a08\u7b97\u3092\u3059\u308b\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[36]:<\/div>\n
      \n
      \n
      \n
      ndiv<\/span> =<\/span> 20<\/span>\r\nNdiv<\/span> =<\/span> 40<\/span>\r\nN<\/span> =<\/span> ndiv<\/span> *<\/span> Ndiv<\/span>\r\n\r\nT2<\/span>,<\/span> X2<\/span>,<\/span> V2<\/span> =<\/span> rk2<\/span>(<\/span>F<\/span>,<\/span> x0<\/span>,<\/span> v0<\/span>,<\/span> t0<\/span>,<\/span> t1<\/span>,<\/span> N<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      T2, X2, V2<\/code> \u306b\u306f N+1<\/code> \u70b9\u306e\u6570\u5024\u89e3\u306e\u5024\u304c\u4ee3\u5165\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u6ed1\u3089\u304b\u306a\u66f2\u7dda\u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u306b\u306f\uff0c\u5168\u90e8\u306e\u70b9\u3092\u4f7f\u3048\u3070\u3044\u3044\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[37]:<\/div>\n
      \n
      \n
      \n
      graphics<\/span>(<\/span>\r\n    list_2d<\/span>(<\/span>T2<\/span>,<\/span> X2<\/span>)<\/span>\r\n)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      <\/div>\n
      \n
      <\/p>\n
      \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u3053\u3053\u3067\u306f\u5c11\u3057\u9593\u5f15\u3044\u3066\u30b0\u30e9\u30d5\u306b\u3057\u3066\u307f\u307e\u3059\u3002\uff08\u3042\u3068\u3067\u89e3\u6790\u89e3\u3068\u6bd4\u3079\u3066\u307f\u308b\u305f\u3081\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[38]:<\/div>\n
      \n
      \n
      \n
      # \u30c7\u30fc\u30bf\u3092\u9593\u5f15\u304f<\/span>\r\nt2<\/span> =<\/span> [<\/span>T2<\/span>[<\/span>i<\/span>]<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>0<\/span>,<\/span> len<\/span>(<\/span>T2<\/span>),<\/span> ndiv<\/span>)]<\/span>\r\nx2<\/span> =<\/span> [<\/span>X2<\/span>[<\/span>i<\/span>]<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>0<\/span>,<\/span> len<\/span>(<\/span>X2<\/span>),<\/span> ndiv<\/span>)]<\/span>\r\n\r\nprint<\/span>(<\/span>'\u5143\u306e\u8981\u7d20\u6570\u306f    '<\/span>,<\/span>len<\/span>(<\/span>X2<\/span>))<\/span>\r\nprint<\/span>(<\/span>'<\/span>%2d<\/span> \u98db\u3073\u306b\u9593\u5f15\u304f\u3068 <\/span>%2d<\/span>'<\/span> %<\/span> (<\/span>ndiv<\/span>,<\/span> len<\/span>(<\/span>x2<\/span>)))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      <\/div>\n
      \n
      \u5143\u306e\u8981\u7d20\u6570\u306f     801\r\n20 \u98db\u3073\u306b\u9593\u5f15\u304f\u3068 41\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[39]:<\/div>\n
      \n
      \n
      \n
      graphics<\/span>(<\/span>\r\n    list_2d<\/span>(<\/span>t2<\/span>,<\/span> x2<\/span>,<\/span> scatter<\/span> =<\/span> True<\/span>,<\/span> \r\n            # \"ms\" \u3064\u307e\u308a \"markersize\" \u3092\u5c0f\u3055\u3081\u306b<\/span>\r\n            rendering_kw<\/span>=<\/span>{<\/span>\"ms\"<\/span>:<\/span>3<\/span>}),<\/span> \r\n    xlabel<\/span> =<\/span> \"$t$\"<\/span>,<\/span> ylabel<\/span> =<\/span> \"$x(t)$\"<\/span>,<\/span>\r\n    title<\/span> =<\/span> \"\u6e1b\u8870\uff0b\u5f37\u5236\u632f\u52d5\"<\/span>\r\n)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      <\/div>\n
      \n
      <\/p>\n
      \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u53c2\u8003\uff1aSymPy \u306b\u3088\u308b2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u89e3\u6790\u89e3<\/h4>\n
      \u7c21\u5358\u306a\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5834\u5408\u306f SymPy \u306e dsolve()<\/code> \u3067\u89e3\u6790\u7684\u306b\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002
      \n\u4f8b\u3068\u3057\u3066\uff0c\u7c21\u5358\u306a\u6e1b\u8870\uff0b\u5f37\u5236\u632f\u52d5\u306e\u65b9\u7a0b\u5f0f<\/p>\n
      $$ \\frac{d^2 x}{dt^2} = -x -a \\frac{dx}{dt} + b \\cos(t) $$<\/p>\n
      \u3092\uff0c<\/p>\n
        \n
      • $a , b $ \u306b\u3044\u304f\u3064\u304b\u5024\u3092\u5165\u308c\u3066<\/li>\n
      • \u521d\u671f\u6761\u4ef6 $t_0 = 0$ \u3067 $x_0 = x(t_0) = 3, v_0 = v(t_0) = 0$\uff0c
        \n\u3068\u3057\u3066\u89e3\u304d\u307e\u3059\u3002<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        <\/div>\n
        \n
        \n
        \u307e\u305a\uff0c$a, b$ \u3092\u5909\u6570\u3068\u3057\u305f\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u8fd4\u3059\u95a2\u6570 eq(a, b)<\/code> \u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        \n
        In\u00a0[40]:<\/div>\n
        \n
        \n
        \n
        var<\/span>(<\/span>'t a b'<\/span>)<\/span>\r\nx<\/span> =<\/span> Function<\/span>(<\/span>'x'<\/span>)(<\/span>t<\/span>)<\/span>\r\n\r\ndef<\/span> eq<\/span>(<\/span>a<\/span>,<\/span> b<\/span>):<\/span>\r\n    return<\/span> Eq<\/span>(<\/span>Derivative<\/span>(<\/span>x<\/span>,<\/span> t<\/span>,<\/span> 2<\/span>),<\/span> \r\n              -<\/span>x<\/span> -<\/span>a<\/span>*<\/span>Derivative<\/span>(<\/span>x<\/span>,<\/span> t<\/span>)<\/span> +<\/span> b<\/span>*<\/span>cos<\/span>(<\/span>t<\/span>))<\/span>\r\n\r\neq<\/span>(<\/span>a<\/span>,<\/span> b<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        \n
        \n
        Out[40]:<\/div>\n
        $\\displaystyle \\frac{d^{2}}{d t^{2}} x{\\left(t \\right)} = -a \\frac{d}{d t} x{\\left(t \\right)} + b \\cos{\\left(t \\right)} -x{\\left(t \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        <\/div>\n
        \n
        \n
        \u307e\u305a\u306f $a = 0, b = 0$ \u306e\u5834\u5408\u306b\uff0c\u521d\u671f\u6761\u4ef6 $x(0) = 3, v(0) = \\dot{x}(0) = 0$ \u306e\u3082\u3068\u3067 dsolve()<\/code> \u3067\u89e3\u6790\u7684\u306b\u89e3\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        \n
        In\u00a0[41]:<\/div>\n
        \n
        \n
        \n
        dsolve<\/span>(<\/span>eq<\/span>(<\/span>0<\/span>,<\/span> 0<\/span>),<\/span> x<\/span>,<\/span> \r\n       ics<\/span> =<\/span> {<\/span>x<\/span>.<\/span>subs<\/span>(<\/span>t<\/span>,<\/span> 0<\/span>):<\/span>3<\/span>,<\/span> diff<\/span>(<\/span>x<\/span>,<\/span> t<\/span>)<\/span>.<\/span>subs<\/span>(<\/span>t<\/span>,<\/span>0<\/span>):<\/span>0<\/span>})<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        \n
        \n
        Out[41]:<\/div>\n
        $\\displaystyle x{\\left(t \\right)} = 3 \\cos{\\left(t \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        <\/div>\n
        \n
        \n
        \u6b21\u306b\uff0c$a = \\frac{1}{2}, b = \\frac{1}{5}$ \u306e\u5834\u5408\u3002\u4ee5\u4e0b\u306e\u3088\u3046\u306b Rational()<\/code> \u95a2\u6570\u3092\u4f7f\u3046\u3068\uff0c\u5b9f\u6570\u5024\u306b\u5909\u63db\u3057\u306a\u3044\u3067\uff08\u7d04\u5206\u3055\u308c\u305f\uff09\u5206\u6570\u306e\u307e\u307e\u3067\u6271\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        \n
        In\u00a0[42]:<\/div>\n
        \n
        \n
        \n
        a<\/span> =<\/span> Rational<\/span>(<\/span>1<\/span>,<\/span>2<\/span>)<\/span>\r\nb<\/span> =<\/span> Rational<\/span>(<\/span>1<\/span>,<\/span>5<\/span>)<\/span>\r\nsol2<\/span> =<\/span> dsolve<\/span>(<\/span>eq<\/span>(<\/span>a<\/span>,<\/span> b<\/span>),<\/span> x<\/span>,<\/span> \r\n              ics<\/span> =<\/span> {<\/span>x<\/span>.<\/span>subs<\/span>(<\/span>t<\/span>,<\/span> 0<\/span>):<\/span>3<\/span>,<\/span> diff<\/span>(<\/span>x<\/span>,<\/span> t<\/span>)<\/span>.<\/span>subs<\/span>(<\/span>t<\/span>,<\/span>0<\/span>):<\/span>0<\/span>})<\/span>\r\nsol2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        \n
        \n
        Out[42]:<\/div>\n
        $\\displaystyle x{\\left(t \\right)} = \\left(\\frac{7 \\sqrt{15} \\sin{\\left(\\frac{\\sqrt{15} t}{4} \\right)}}{75} + 3 \\cos{\\left(\\frac{\\sqrt{15} t}{4} \\right)}\\right) e^{-\\frac{t}{4}} + \\frac{2 \\sin{\\left(t \\right)}}{5}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        <\/div>\n
        \n
        \n
        \u6c42\u307e\u3063\u305f\u89e3\u6790\u89e3\u3092\u95a2\u6570\u3068\u3057\u3066\u5b9a\u7fa9\u3059\u308b\u4f8b\u3067\u3059\u3002\uff08\u3061\u3087\u3063\u3068\u30c8\u30ea\u30c3\u30ad\u30fc\u3067\u3059\u304b\u306d\uff1f\uff09<\/p>\n
        \u89e3 sol2<\/code> \u306e\u53f3\u8fba rhs<\/code> \u306e t<\/code> \u306b s<\/code> \u3092\u4ee3\u5165\u3057\u3066\u95a2\u6570 xsol2(s)<\/code> \u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        \n
        In\u00a0[43]:<\/div>\n
        \n
        \n
        \n
        def<\/span> xsol2<\/span>(<\/span>s<\/span>):<\/span>\r\n    return<\/span> sol2<\/span>.<\/span>rhs<\/span>.<\/span>subs<\/span>(<\/span>t<\/span>,<\/span> s<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        \n
        In\u00a0[44]:<\/div>\n
        \n
        \n
        \n
        key1<\/span> =<\/span> \"$a = <\/span>%3.1f<\/span>, b = <\/span>%3.1f<\/span>$ \u306e\u89e3\u6790\u89e3\"<\/span> %<\/span> (<\/span>a<\/span>,<\/span> b<\/span>)<\/span>\r\nkey2<\/span> =<\/span> \"$a = <\/span>%3.1f<\/span>, b = <\/span>%3.1f<\/span>$ \u306e\u6570\u5024\u89e3\"<\/span> %<\/span> (<\/span>a<\/span>,<\/span> b<\/span>)<\/span>\r\n\r\ngraphics<\/span>(<\/span>\r\n    line<\/span>(<\/span>xsol2<\/span>(<\/span>t<\/span>),<\/span> (<\/span>t<\/span>,<\/span> 0<\/span>,<\/span> 20<\/span>),<\/span> key1<\/span>),<\/span> \r\n    list_2d<\/span>(<\/span>t2<\/span>,<\/span> x2<\/span>,<\/span> key2<\/span>,<\/span> scatter<\/span> =<\/span> True<\/span>,<\/span> \r\n            rendering_kw<\/span>=<\/span>{<\/span>\"ms\"<\/span>:<\/span>3<\/span>}),<\/span> \r\n    xlim<\/span> =<\/span> (<\/span>0<\/span>,<\/span> 20<\/span>),<\/span> \r\n    xlabel<\/span> =<\/span> \"$t$\"<\/span>,<\/span> ylabel<\/span> =<\/span> \"$x(t)$\"<\/span>,<\/span>\r\n    title<\/span> =<\/span> \"\u6e1b\u8870\uff0b\u5f37\u5236\u632f\u52d5\"<\/span>\r\n)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        \n
        \n
        <\/div>\n
        \n
        <\/p>\n
        \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        <\/div>\n
        \n
        \n
        \u25cb\u7df4\u7fd2\uff1a\u6e1b\u8870\uff0b\u5f37\u5236\u632f\u52d5<\/h4>\n
        \u6e1b\u8870\uff0b\u5f37\u5236\u632f\u52d5\u306e\u65b9\u7a0b\u5f0f\u3092\u4ee5\u4e0b\u306e\u5834\u5408\u306b\u89e3\u304d\uff0c\u30b0\u30e9\u30d5\u306b\u3059\u308b\u3002<\/p>\n
        \u307e\u305a $a$ \u306e\u5f71\u97ff\u3092\u8abf\u3079\u308b\u305f\u3081\uff0c<\/p>\n
          \n
        1. $a = 0, b = 0$<\/li>\n
        2. $a = 0.2, b = 0$<\/li>\n
        3. $a = 0.5, b = 0$<\/li>\n
        4. $a = 0.8, b = 0$<\/li>\n<\/ol>\n
          \u6b21\u306b\uff0c$b$ \u306e\u5f71\u97ff\u3092\u8abf\u3079\u308b\u305f\u3081\uff0c<\/p>\n
            \n
          1. $a = 0, b = 0$<\/li>\n
          2. $a = 0, b = 0.2$<\/li>\n
          3. $a = 0, b = 0.5$<\/li>\n
          4. $a = 0, b = 0.8$<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n
            \n
            <\/div>\n
            \n
            \n
            \u25cb\u7df4\u7fd2\uff1a\u5358\u632f\u308a\u5b50<\/h4>\n
            2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/p>\n
            $$ \\frac{d^2\\theta}{dt^2} = -4\\pi^2 \\sin\\theta$$<\/p>\n
            \u3092\u521d\u671f\u6761\u4ef6<\/p>\n
            $$ \\theta = 45^{\\circ}, \\ \\ \\frac{d\\theta}{dt} = 0 \\quad \\mbox{at}\\ \\ t = 0$$<\/p>\n
            \u306e\u3082\u3068\u3067 $t=0$ \u304b\u3089 $t = 3$ \u307e\u3067\u6570\u5024\u7684\u306b\u89e3\u3051\u3002\uff08\u89e3\u6790\u7684\u306b\u306f\u89e3\u3051\u307e\u305b\u3093\u3088\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
            \n
            <\/div>\n
            \n
            \n
            $\\theta \\Rightarrow x, \\frac{d\\theta}{dt} \\Rightarrow v$ \u3068\u7f6e\u304d\u63db\u3048\u3066\uff0crk2()<\/code> \u3092\u4f7f\u3046\u3002\u521d\u671f\u6761\u4ef6\u306e\u89d2\u5ea6\u306f\u30e9\u30b8\u30a2\u30f3\u306b\u3057\u3066 $\\sin\\theta$ \u306b\u6e21\u3059\u3088\u3046\u306b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
            \n
            \n
            In\u00a0[45]:<\/div>\n
            \n
            \n
            \n
            # \u5909\u6570 t, x, v \u306e\u521d\u671f\u5316<\/span>\r\nvar<\/span>(<\/span>'t x v'<\/span>)<\/span>\r\n\r\ndef<\/span> F<\/span>(<\/span>t<\/span>,<\/span> x<\/span>,<\/span> v<\/span>):<\/span>\r\n    # pi \u306e\u307e\u307e\u3060\u3068\u53b3\u5bc6\u306b\u8a08\u7b97\u3057\u3088\u3046\u3068\u3057\u3066<\/span>\r\n    # \u3082\u306e\u3059\u3054\u304f\u6642\u9593\u304c\u304b\u304b\u3063\u3066\u3057\u307e\u3046\u306e\u3067<\/span>\r\n    return<\/span> -<\/span>4<\/span>*<\/span>float<\/span>(<\/span>pi<\/span>)<\/span>**<\/span>2<\/span>*<\/span>sin<\/span>(<\/span>x<\/span>)<\/span>\r\n\r\nx0<\/span> =<\/span> float<\/span>(<\/span>rad<\/span>(<\/span>45<\/span>))<\/span>\r\nv0<\/span> =<\/span> 0<\/span>\r\nt0<\/span> =<\/span> 0<\/span>\r\nt1<\/span> =<\/span> 3<\/span>\r\n\r\nfor<\/span> N<\/span> in<\/span> [<\/span>100<\/span>,<\/span> 200<\/span>,<\/span> 400<\/span>,<\/span> 800<\/span>,<\/span> 1600<\/span>,<\/span> 3200<\/span>]:<\/span>\r\n    T2<\/span>,<\/span> X2<\/span>,<\/span> V2<\/span> =<\/span> rk2<\/span>(<\/span>F<\/span>,<\/span> x0<\/span>,<\/span> v0<\/span>,<\/span> t0<\/span>,<\/span> t1<\/span>,<\/span> N<\/span>)<\/span>\r\n    # X2 \u306e\u6700\u5f8c\u306e\u8981\u7d20 X2[-1] \u3092\u8868\u793a<\/span>\r\n    print<\/span>(<\/span>\"N =<\/span>%5d<\/span>, x(t1) =<\/span>%15.12f<\/span>\"<\/span> %<\/span> (<\/span>N<\/span>,<\/span> X2<\/span>[<\/span>-<\/span>1<\/span>]))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
            \n
            \n
            \n
            <\/div>\n
            \n
            N =  100, x(t1) = 0.591465894161\r\nN =  200, x(t1) = 0.591544091180\r\nN =  400, x(t1) = 0.591548830281\r\nN =  800, x(t1) = 0.591549121191\r\nN = 1600, x(t1) = 0.591549139198\r\nN = 3200, x(t1) = 0.591549140318\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
            \n
            <\/div>\n
            \n
            \n
            \u4e0a\u306e\u7d50\u679c\u304b\u3089\uff0cN = 1600<\/code> \u3067\u5c0f\u6570\u70b9\u4ee5\u4e0b 7~8 \u6841\u7a0b\u5ea6\u306e\u7cbe\u5ea6\u304c\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n
            \u4ee5\u4e0b\u306e\u3088\u3046\u306b\u7d50\u69cb\u6642\u9593\u304c\u304b\u304b\u308b\u304c\uff0c\u3053\u308c\u3067\u884c\u3053\u3046\u3002<\/p>\n
            \u30bb\u30eb\u306e\u6700\u521d\u306b %%time<\/code> \u3068\u66f8\u304f\u3053\u3068\u3067\uff0c\u3053\u306e\u30bb\u30eb\u3092\u5b9f\u884c\u3059\u308b\u306e\u306b\u304b\u304b\u3063\u305f\u6642\u9593\u304c\u308f\u304b\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
            \n
            \n
            In\u00a0[46]:<\/div>\n
            \n
            \n
            \n
            %%time<\/span>\r\n\r\nT2<\/span>,<\/span> X2<\/span>,<\/span> V2<\/span> =<\/span> rk2<\/span>(<\/span>F<\/span>,<\/span> x0<\/span>,<\/span> v0<\/span>,<\/span> t0<\/span>,<\/span> t1<\/span>,<\/span> 1600<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
            \n
            \n
            \n
            <\/div>\n
            \n
            CPU times: user 6.91 s, sys: 0 ns, total: 6.91 s\r\nWall time: 6.91 s\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
            \n
            <\/div>\n
            \n
            \n
            \u4e00\u65b9\uff0c$\\theta$ \u304c\u3042\u307e\u308a\u5927\u304d\u304f\u306a\u3044\u3068\u3059\u308c\u3070\uff0c\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306e1\u6b21\u306e\u8fd1\u4f3c\u3067 $\\sin\\theta \\simeq \\theta$ \u3067\u3042\u308b\u304b\u3089<\/p>\n
            $$ \\frac{d^2\\theta}{dt^2} \\simeq -4\\pi^2 \\theta$$<\/p>\n
            $\\simeq$ \u3092 $=$ \u306b\u5909\u3048\u3066\u65b9\u7a0b\u5f0f\u306b\u3059\u308b\u3068\uff0c\u3053\u308c\u306f\u5358\u632f\u52d5\u306e\u65b9\u7a0b\u5f0f\u3067\u3042\u308b\u304b\u3089\uff0c\u540c\u3058\u521d\u671f\u6761\u4ef6\u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u89e3\u6790\u7684\u306b\u89e3\u3051\u308b\u3002<\/p>\n
            $$\\theta_s = \\theta_0 \\cos (2 \\pi t) = \\frac{\\pi}{4} \\cos (2 \\pi t)$$<\/p>\n
            \u3053\u306e\u5358\u632f\u52d5\u306e\u89e3\u3082\u91cd\u306d\u3066\u30b0\u30e9\u30d5\u306b\u3057\u3066\u307f\u3088\u3002\u5468\u671f\u306b\u306f\u3069\u306e\u3088\u3046\u306a\u9055\u3044\u304c\u3042\u308b\u304b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
            \n
            \n
            In\u00a0[47]:<\/div>\n
            \n
            \n
            \n
            # \u7e26\u8ef8\u306e\u30ea\u30b9\u30c8 X2 \u306e\u5404\u8981\u7d20\u3092\u30e9\u30b8\u30a2\u30f3\u304b\u3089\u5ea6\u3078\u5909\u63db<\/span>\r\nXX2<\/span> =<\/span> [<\/span>deg<\/span>(<\/span>1<\/span>)<\/span>*<\/span>X2<\/span>[<\/span>i<\/span>]<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>len<\/span>(<\/span>X2<\/span>))]<\/span>\r\n\r\ngraphics<\/span>(<\/span>\r\n    list_2d<\/span>(<\/span>T2<\/span>,<\/span> XX2<\/span>,<\/span> '\u6570\u5024\u89e3'<\/span>),<\/span> \r\n    line<\/span>(<\/span>deg<\/span>(<\/span>pi<\/span>\/<\/span>4<\/span>)<\/span> *<\/span> cos<\/span>(<\/span>2<\/span>*<\/span>pi<\/span>*<\/span>t<\/span>),<\/span> (<\/span>t<\/span>,<\/span> 0<\/span>,<\/span> 3<\/span>),<\/span> '\u5358\u632f\u52d5\u89e3'<\/span>),<\/span> \r\n    xlabel<\/span> =<\/span> '$t$'<\/span>,<\/span> ylabel<\/span> =<\/span> r<\/span>'$\\theta$ (\u5ea6)'<\/span>\r\n)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
            \n
            \n
            \n
            <\/div>\n
            \n
            <\/p>\n
            \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
            \n
            <\/div>\n
            \n
            \n
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            \u307e\u305f SymPy \u81ea\u4f53\u3067\u3082\u30b0\u30e9\u30d5\u4f5c\u6210\u6a5f\u80fd\u304c\u3042\u308a\u307e\u3059\u304c\uff0c\u3053\u3053\u3067\u306f\uff0cSymPy Plotting Backends (SPB) \u3092\u4f7f\u3063\u3066\u30b0\u30e9\u30d5\u3092\u63cf\u3044\u3066\u307f\u307e\u3059\u3002<\/p>
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