{"id":9659,"date":"2024-11-13T22:00:12","date_gmt":"2024-11-13T13:00:12","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=9659"},"modified":"2024-11-14T10:12:21","modified_gmt":"2024-11-14T01:12:21","slug":"%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\/sympy-%e6%bc%94%e7%bf%92\/%e5%b8%b8%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%bc%8f\/","title":{"rendered":"4. SymPy \u3067\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f"},"content":{"rendered":"

SymPy \u30671\u968e\u7dda\u5f62\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u30842\u968e\u5b9a\u6570\u4fc2\u6570\u7dda\u5f62\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u6790\u7684\u306b\u89e3\u304f\u3002<\/p>\n

\n
<\/div>\n
\n
\n

SymPy \u306e import<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[1]:<\/div>\n
\n
\n
\n
from<\/span> sympy.abc<\/span> import<\/span> *<\/span>\r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h3>\n
\u672a\u5b9a\u7fa9\u95a2\u6570\u306e\u5ba3\u8a00 Function()<\/code><\/h4>\n
$y$ \u3092\uff08\u5909\u6570\u3067\u306f\u306a\u304f\uff09\u95a2\u6570\u3068\u3057\u3066\u5ba3\u8a00\u3059\u308b\u306b\u306f y = Function('y')<\/code> \u3068\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[2]:<\/div>\n
\n
\n
\n
y<\/span> =<\/span> Function<\/span>(<\/span>'y'<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u4e0a\u8a18\u306e\u3088\u3046\u306b\u95a2\u6570\u3068\u3057\u3066\u5ba3\u8a00\u3059\u308b\u3068 y<\/code> \u306e\u30bf\u30a4\u30d7\u306f UndefinedFunction<\/code> \u672a\u5b9a\u7fa9\u95a2\u6570\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[3]:<\/div>\n
\n
\n
\n
type<\/span>(<\/span>y<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[3]:<\/div>\n
\n
sympy.core.function.UndefinedFunction<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u672a\u5b9a\u7fa9\u95a2\u6570\u3068\u3057\u3066\u5ba3\u8a00\u3055\u308c\u305f $y$ \u3092\u5909\u6570 $x$ \u306e\u95a2\u6570 $y(x)$ \u3068\u3059\u308b\u5834\u5408\u306f\uff0cy(x)<\/code> \u3068\u66f8\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[4]:<\/div>\n
\n
\n
\n
y<\/span>(<\/span>x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[4]:<\/div>\n
$\\displaystyle y{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u53c2\u8003\uff1a\u8907\u6570\u306e\u672a\u5b9a\u7fa9\u95a2\u6570\u306e\u5ba3\u8a00<\/h5>\n
\u8907\u6570\u306e\u672a\u5b9a\u7fa9\u95a2\u6570\u3092\u5ba3\u8a00\u3059\u308b\u969b\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u66f8\u304d\u65b9\u3082\u3067\u304d\u308b\u3088\u3046\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[5]:<\/div>\n
\n
\n
\n
# f, g, h = Function('f g h') \u306a\u3069\u3068\u306f\u66f8\u3051\u306a\u3044\u3002<\/span>\r\n# <\/span>\r\n# f = Function('f')<\/span>\r\n# g = Function('g')<\/span>\r\n# h = Function('h') \u30683\u884c\u66f8\u304f\u3088\u308a\u3082\u7c21\u5358<\/span>\r\n\r\nvar<\/span>(<\/span>'f g h'<\/span>,<\/span> cls<\/span> =<\/span> Function<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[5]:<\/div>\n
\n
(f, g, h)<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u5fae\u5206\u306e\u8868\u8a18 Derivative()<\/code>, diff()<\/code><\/h4>\n
\u5fae\u5206 $\\displaystyle \\frac{dy}{dx}$ \u3092\u8868\u3059\u306b\u306f Derivative(y(x), x)<\/code> \u307e\u305f\u306f diff(y(x), x)<\/code>\u3002\u672a\u5b9a\u7fa9\u95a2\u6570\u306b\u3064\u3044\u3066\u306f diff(y(x), x)<\/code> \u306f Derivative(y(x), x)<\/code> \u3068\u540c\u3058\u3067\u3059\u3002<\/p>\n
$\\displaystyle \\frac{dy}{dx} = $ Derivative(y(x), x)<\/code> $=$ diff(y(x), x)<\/code> $=$ y(x).diff(x)<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[6]:<\/div>\n
\n
\n
\n
diff<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[6]:<\/div>\n
$\\displaystyle \\frac{d}{d x} y{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[7]:<\/div>\n
\n
\n
\n
# \u3053\u306e\u3088\u3046\u306a\u66f8\u304d\u65b9\u3082\u3067\u304d\u308b<\/span>\r\n\r\ny<\/span>(<\/span>x<\/span>)<\/span>.<\/span>diff<\/span>(<\/span>x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[7]:<\/div>\n
$\\displaystyle \\frac{d}{d x} y{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[8]:<\/div>\n
\n
\n
\n
# \u672a\u5b9a\u7fa9\u95a2\u6570 y \u3092 t \u306e\u95a2\u6570\u3068\u3057\u305f\u3044\u306a\u3089...<\/span>\r\ndisplay<\/span>(<\/span>y<\/span>(<\/span>t<\/span>))<\/span>\r\ndisplay<\/span>(<\/span>y<\/span>(<\/span>t<\/span>)<\/span>.<\/span>diff<\/span>(<\/span>t<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
$\\displaystyle y{\\left(t \\right)}$<\/div>\n<\/div>\n
\n
<\/div>\n
$\\displaystyle \\frac{d}{d t} y{\\left(t \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5b9a\u7fa9 Eq()<\/code><\/h4>\n
\u4f8b\u3068\u3057\u3066\u5fae\u5206\u65b9\u7a0b\u5f0f $\\displaystyle \\frac{dy}{dx} = -2 x\\, y$ \u306e\u5b9a\u7fa9\u306e\u4ed5\u65b9\u306b\u3064\u3044\u3066\u307e\u3068\u3081\u307e\u3059\u3002<\/p>\n
\u65b9\u7a0b\u5f0f\u3068\u306f\u672a\u77e5\u5909\u6570\u3084\u672a\u77e5\u95a2\u6570\u3092\u542b\u3080\u7b49\u5f0f\u306e\u3053\u3068\u3067\u3059\u304b\u3089\uff0c\u7b49\u5f0f\u3068\u540c\u3058\u3088\u3046\u306b
\nEq(\u5de6\u8fba, \u53f3\u8fba)<\/code> \u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[9]:<\/div>\n
\n
\n
\n
# y \u3092\u95a2\u6570\u3068\u3057\u3066\u5ba3\u8a00<\/span>\r\ny<\/span> =<\/span> Function<\/span>(<\/span>'y'<\/span>)<\/span>\r\n\r\n# \u89e3\u304f\u3079\u304d\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u5b9a\u7fa9\u3057\u3066 eq \u306b\u4ee3\u5165<\/span>\r\neq<\/span> =<\/span> Eq<\/span>(<\/span>diff<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> -<\/span>2<\/span>*<\/span>x<\/span>*<\/span>y<\/span>(<\/span>x<\/span>))<\/span>\r\n\r\n# eq \u306b\u4ee3\u5165\u3055\u308c\u305f\u65b9\u7a0b\u5f0f\u3092\u8868\u793a<\/span>\r\neq<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[9]:<\/div>\n
$\\displaystyle \\frac{d}{d x} y{\\left(x \\right)} = – 2 x y{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f dsolve()<\/code><\/h4>\n
SymPy \u3067\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u306b\u306f dsolve()<\/code> \u95a2\u6570\u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n
eq<\/code> \u306b\u4ee3\u5165\u3055\u308c\u305f\u5fae\u5206\u65b9\u7a0b\u5f0f
\n$\\displaystyle \\frac{dy}{dx} = -2 x\\, y$ \u3092\u672a\u77e5\u95a2\u6570 y(x)<\/code> \u306b\u3064\u3044\u3066\u89e3\u304f\u306b\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[10]:<\/div>\n
\n
\n
\n
# dsolve() \u3067\u5fae\u5206\u65b9\u7a0b\u5f0f eq \u3092 y(x) \u306b\u3064\u3044\u3066\u89e3\u304f<\/span>\r\n\r\ndsolve<\/span>(<\/span>eq<\/span>,<\/span> y<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[10]:<\/div>\n
$\\displaystyle y{\\left(x \\right)} = C_{1} e^{- x^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u4e0a\u8a18\u306e\u51fa\u529b\u7d50\u679c\u306b\u73fe\u308c\u308b $C_1$ \u306f\u7a4d\u5206\u5b9a\u6570\u3067\u3059\u3002<\/p>\n
\u3061\u306a\u307f\u306b\uff0c\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c<\/p>\n
$$\\frac{dy}{dx} + 2 x y = 0$$<\/p>\n
\u306e\u3088\u3046\u306b\u53f3\u8fba\u304c\u30bc\u30ed\u306e\u5f62\u306b\u306a\u3063\u3066\u3044\u308b\u5834\u5408\u306f\uff0c\u5de6\u8fba\u306e\u307f\u3092 dsolve()<\/code> \u306e\u5f15\u6570\u306b\u3057\u3066\u3082\u540c\u3058\u7b54\u3048\u304c\u51fa\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[11]:<\/div>\n
\n
\n
\n
sahen<\/span> =<\/span> diff<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span> +<\/span> 2<\/span>*<\/span>x<\/span>*<\/span>y<\/span>(<\/span>x<\/span>)<\/span>\r\ndisplay<\/span>(<\/span>sahen<\/span>)<\/span>\r\n\r\ndsolve<\/span>(<\/span>sahen<\/span>,<\/span> y<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
$\\displaystyle 2 x y{\\left(x \\right)} + \\frac{d}{d x} y{\\left(x \\right)}$<\/div>\n<\/div>\n
\n
Out[11]:<\/div>\n
$\\displaystyle y{\\left(x \\right)} = C_{1} e^{- x^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u521d\u671f\u6761\u4ef6 ics={}<\/code><\/h5>\n
\u521d\u671f\u6761\u4ef6\u3092\u8ab2\u3057\u3066\u7a4d\u5206\u5b9a\u6570\u306e\u5024\u3092\u6c7a\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n
\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u969b\u306e\u521d\u671f\u6761\u4ef6\u306f dsolve()<\/code> \u306e\u30aa\u30d7\u30b7\u30e7\u30f3\u306b ics={}<\/code> \u306e\u3088\u3046\u306b\u300c\u8f9e\u66f8\u300d\u5f62\u5f0f\u3067\u8a2d\u5b9a\u3057\u307e\u3059\u3002<\/p>\n
\u4f8b\u3068\u3057\u3066<\/p>\n
$$\\frac{dy}{dx} + 2 x y = 0$$<\/p>\n
\u3092 $x = 0$ \u3067 $y(0) = 1$ \u3068\u3044\u3046\u521d\u671f\u6761\u4ef6\u306e\u3082\u3068\u3067\u89e3\u304f\u5834\u5408\uff1a<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[12]:<\/div>\n
\n
\n
\n
dsolve<\/span>(<\/span>sahen<\/span>,<\/span> y<\/span>(<\/span>x<\/span>),<\/span> ics<\/span>=<\/span>{<\/span>y<\/span>(<\/span>0<\/span>):<\/span>1<\/span>})<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[12]:<\/div>\n
$\\displaystyle y{\\left(x \\right)} = e^{- x^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u306e\u521d\u671f\u6761\u4ef6\u306b\u3088\u3063\u3066\u7a4d\u5206\u5b9a\u6570\u304c $C_1 = 1$ \u3068\u6c7a\u307e\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
1\u968e\u7dda\u5f62\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h4>\n
\u3044\u304f\u3064\u304b\u4f8b\u984c\u3092\u89e3\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n
\u30de\u30eb\u30b5\u30b9\u306e\u4eba\u53e3\u30e2\u30c7\u30eb<\/h5>\n
\u3042\u308b\u6642\u523b $t$ \u306b\u304a\u3051\u308b\uff0c\u3068\u3042\u308b\u56fd\uff08\u5730\u57df\uff09\u306e\u7dcf\u4eba\u53e3\u3092 $N(t)$ \u3068\u3059\u308b\u3068\uff0c\u30de\u30eb\u30b5\u30b9\u306e\u4eba\u53e3\u30e2\u30c7\u30eb\u306f\u4ee5\u4e0b\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u8868\u3055\u308c\u307e\u3059\u3002<\/p>\n
$$\\frac{dN}{dt} = \\gamma\\, N$$<\/p>\n
\u3053\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u521d\u671f\u6761\u4ef6 $t=t_0$ \u3067 $N(t_0) = N_0$ \u306e\u3082\u3068\u3067\u89e3\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[13]:<\/div>\n
\n
\n
\n
# N \u3092\u95a2\u6570\u3068\u3057\u3066\u5ba3\u8a00<\/span>\r\nN<\/span> =<\/span> Function<\/span>(<\/span>'N'<\/span>)<\/span>\r\n# gamma \u3092\u5909\u6570\u3068\u3057\u3066\u5ba3\u8a00\uff08\u30ac\u30f3\u30de\u95a2\u6570\u3068\u30ab\u30d6\u308b\u305f\u3081\uff09<\/span>\r\nvar<\/span>(<\/span>'gamma'<\/span>)<\/span>\r\n\r\n# \u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5b9a\u7fa9<\/span>\r\neqm<\/span> =<\/span> Eq<\/span>(<\/span>diff<\/span>(<\/span>N<\/span>(<\/span>t<\/span>),<\/span> t<\/span>),<\/span> gamma<\/span>*<\/span>N<\/span>(<\/span>t<\/span>))<\/span>\r\neqm<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[13]:<\/div>\n
$\\displaystyle \\frac{d}{d t} N{\\left(t \\right)} = \\gamma N{\\left(t \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[14]:<\/div>\n
\n
\n
\n
# \u521d\u671f\u5024\u5909\u6570\u306e\u5ba3\u8a00<\/span>\r\nvar<\/span>(<\/span>'t0, N0'<\/span>)<\/span>\r\n\r\n# \u521d\u671f\u6761\u4ef6\u306f ics={} \u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b...<\/span>\r\nsolm<\/span> =<\/span> dsolve<\/span>(<\/span>eqm<\/span>,<\/span> N<\/span>(<\/span>t<\/span>),<\/span> ics<\/span>=<\/span>{<\/span>N<\/span>(<\/span>t0<\/span>):<\/span>N0<\/span>})<\/span>\r\nsolm<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[14]:<\/div>\n
$\\displaystyle N{\\left(t \\right)} = N_{0} e^{\\gamma t} e^{- \\gamma t_{0}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30f4\u30a7\u30a2\u30d5\u30eb\u30b9\u30c8\u306b\u3088\u308b\u4fee\u6b63\u4eba\u53e3\u30e2\u30c7\u30eb<\/h5>\n
\u30de\u30eb\u30b5\u30b9\u306e\u4eba\u53e3\u30e2\u30c7\u30eb\u306f\uff0c$\\gamma > 0$ \u306e\u5834\u5408\u306b\u306f\u4eba\u53e3\u306e\u969b\u9650\u306a\u304d\u5897\u52a0\uff08\u307e\u3055\u306b\u6307\u6570\u95a2\u6570\u7684\u306a\u5897\u52a0\uff09\u3092\u4e88\u6e2c\u3059\u308b\u3002\u3057\u304b\u3057\uff0c\u5b9f\u969b\u306b\u306f\u3055\u307e\u3056\u307e\u306a\u8981\u56e0\u306b\u3088\u308a\uff0c\u3053\u306e\u3088\u3046\u306a\u7121\u5236\u9650\u306a\u5897\u52a0\u306f\u7d9a\u304b\u306a\u3044\u3002<\/p>\n
\u4eba\u53e3\u904e\u5bc6\u306e\u8981\u56e0\u3092\u8003\u616e\u306b\u3044\u308c\u305f\u4fee\u6b63\u3092\u53d6\u308a\u5165\u308c\u305f\u30f4\u30a7\u30a2\u30d5\u30eb\u30b9\u30c8\u306e\u30e2\u30c7\u30eb\u306f\uff0c\u4ee5\u4e0b\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
$$\\frac{dN}{dt} = \\gamma N \\left(1 \u2013 \\frac{N}{N_{\\rm max}}\\right)$$<\/p>\n
\u3053\u306e\u65b9\u7a0b\u5f0f\u3092\uff0c\u540c\u3058\u521d\u671f\u6761\u4ef6 $t=t_0$ \u3067 $N(t_0) = N_0$ \u306e\u3082\u3068\u3067\u89e3\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[15]:<\/div>\n
\n
\n
\n
N<\/span> =<\/span> Function<\/span>(<\/span>'N'<\/span>)<\/span>\r\nvar<\/span>(<\/span>'gamma N_max'<\/span>)<\/span>\r\n\r\neqv<\/span> =<\/span> Eq<\/span>(<\/span>diff<\/span>(<\/span>N<\/span>(<\/span>t<\/span>),<\/span> t<\/span>),<\/span> gamma<\/span>*<\/span>N<\/span>(<\/span>t<\/span>)<\/span>*<\/span>(<\/span>1<\/span>-<\/span>N<\/span>(<\/span>t<\/span>)<\/span>\/<\/span>N_max<\/span>))<\/span>\r\neqv<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[15]:<\/div>\n
$\\displaystyle \\frac{d}{d t} N{\\left(t \\right)} = \\gamma \\left(1 – \\frac{N{\\left(t \\right)}}{N_{max}}\\right) N{\\left(t \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[16]:<\/div>\n
\n
\n
\n
var<\/span>(<\/span>'t0 N0'<\/span>)<\/span>\r\n\r\nansv<\/span> =<\/span> dsolve<\/span>(<\/span>eqv<\/span>,<\/span> N<\/span>(<\/span>t<\/span>),<\/span> ics<\/span> =<\/span> {<\/span>N<\/span>(<\/span>t0<\/span>):<\/span>N0<\/span>})<\/span>\r\nansv<\/span>.<\/span>simplify<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[16]:<\/div>\n
$\\displaystyle N{\\left(t \\right)} = \\frac{N_{0} N_{max} e^{\\gamma t}}{N_{0} e^{\\gamma t} – \\left(N_{0} – N_{max}\\right) e^{\\gamma t_{0}}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u7a4d\u5206\u56e0\u5b50\u6cd5\u306e\u4f8b\u984c<\/h5>\n
\uff08\u975e\u540c\u6b21\u9805 $Q(x)$ \u304c\u3042\u308b\uff09\u4e00\u822c\u7684\u306a1\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f<\/p>\n
$$\\frac{dy}{dx} + P(x) y = Q(x)$$<\/p>\n
\u3092\u4eba\u529b\u3067\u89e3\u304f\u969b\u306b\u306f\uff0c\u7a4d\u5206\u56e0\u5b50\u6cd5\u3092\u4f7f\u3044\u307e\u3059\u3002\u7d50\u679c\u306e\u307f\u307e\u3068\u3081\u308b\u3068\uff0c<\/p>\n
$$g(x) = \\exp \\left\\{\\int^x P(x\u2019)\\,dx\u2019 \\right\\}$$<\/p>\n
\u3067\u4e0e\u3048\u3089\u308c\u308b\u7a4d\u5206\u56e0\u5b50<\/strong><\/span> $g(x)$ \u3092\u4f7f\u3046\u3068\uff0c\u89e3\u304c<\/p>\n
$$y = \\frac{1}{g(x)} \\left\\{\\int^x g(x\u2019) Q(x\u2019) \\, dx\u2019 + C \\right\\}$$<\/p>\n
\u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\uff0c\u3068\u3044\u3046\u65b9\u6cd5\u3067\u3059\u3002<\/p>\n
\u7a4d\u5206\u56e0\u5b50\u6cd5\u3092\u4f7f\u3063\u3066\u89e3\u304f\u4f8b\u984c\u3068\u3057\u3066\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n
$$\\frac{dy}{dx} + \\frac{y}{x} = \\frac{\\sin x}{x}$$<\/p>\n
SymPy \u306e dsolve()<\/code> \u3092\u4f7f\u3048\u3070\uff0c\u7279\u306b\u7a4d\u5206\u56e0\u5b50\u6cd5\u3092\u610f\u8b58\u3059\u308b\u3053\u3068\u306a\u304f\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[17]:<\/div>\n
\n
\n
\n
y<\/span> =<\/span> Function<\/span>(<\/span>'y'<\/span>)<\/span>\r\n\r\neq<\/span> =<\/span> Eq<\/span>(<\/span>diff<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span> +<\/span> y<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>x<\/span>,<\/span> sin<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>x<\/span>)<\/span>\r\ndisplay<\/span>(<\/span>eq<\/span>)<\/span>\r\n\r\ndsolve<\/span>(<\/span>eq<\/span>,<\/span> y<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
$\\displaystyle \\frac{d}{d x} y{\\left(x \\right)} + \\frac{y{\\left(x \\right)}}{x} = \\frac{\\sin{\\left(x \\right)}}{x}$<\/div>\n<\/div>\n
\n
Out[17]:<\/div>\n
$\\displaystyle y{\\left(x \\right)} = \\frac{C_{1} – \\cos{\\left(x \\right)}}{x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u306e\u4f8b\u984c\u3092\u3042\u3048\u3066\u7a4d\u5206\u56e0\u5b50\u6cd5\u306e\u516c\u5f0f\u306b\u5f93\u3063\u3066\u3084\u3063\u3066\u307f\u307e\u3059\u3002<\/p>\n
\u516c\u5f0f\u306b\u3042\u3066\u306f\u3081\u308b\u3068\uff0c$\\displaystyle P(x) = \\frac{1}{x}, \\ Q(x) = \\frac{\\sin x}{x}$ \u3067\u3059\u304b\u3089\u7a4d\u5206\u56e0\u5b50 $g(x)$ \u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[18]:<\/div>\n
\n
\n
\n
P<\/span> =<\/span> 1<\/span>\/<\/span>x<\/span>\r\nQ<\/span> =<\/span> sin<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>x<\/span>\r\n\r\ng<\/span> =<\/span> exp<\/span>(<\/span>integrate<\/span>(<\/span>P<\/span>,<\/span> x<\/span>))<\/span>\r\ng<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[18]:<\/div>\n
$\\displaystyle x$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u7a4d\u5206\u56e0\u5b50 $g(x)$ \u304c\u4e0a\u8a18\u306e\u3088\u3046\u306b\u6c42\u307e\u3063\u305f\u306e\u3067\uff0c\u89e3\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u25cb\u7df4\u7fd2\uff1a\u7a4d\u5206\u56e0\u5b50\u304b\u3089\u89e3\u3092\u6c42\u3081\u308b<\/h5>\n
\u5fae\u5206\u65b9\u7a0b\u5f0f<\/p>\n
$$\\frac{dy}{dx} + \\frac{y}{x} = \\frac{\\sin x}{x}$$<\/p>\n
\u306e\u7a4d\u5206\u56e0\u5b50 $g(x)$ \u304c\u4e0a\u306e\u4e0a\u306e\u3088\u3046\u306b\u6c42\u3081\u3089\u308c\u305f\u3068\u304d\uff0c\u3053\u308c\u3092\u4f7f\u3063\u3066\u89e3\u3092\u6c42\u3081\u3088\u3002<\/p>\n
\u30d2\u30f3\u30c8\uff1a<\/p>\n
\u89e3\u306f<\/p>\n
$$y = \\frac{1}{g(x)} \\left\\{\\int^x g(x\u2019) Q(x\u2019) \\, dx\u2019 + C \\right\\}$$<\/p>\n
\u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u304b\u3089…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4f8b\uff1a\u5358\u632f\u52d5<\/h4>\n
\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4f8b\u3068\u3057\u3066\uff0c\u529b\u5b66\u306e\u554f\u984c\u3067\u3088\u304f\u51fa\u308b\uff0c\u5358\u632f\u52d5\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u3068\u3057\u3066\u77e5\u3089\u308c\u308b\u5f0f<\/p>\n
$$ \\frac{d^2 x}{dt^2} + \\omega_0^2 \\,x = 0, \\quad (\\omega_0 > 0)$$<\/p>\n
\u3092\u89e3\u3044\u3066\u307f\u307e\u3059\u3002$x(t)$ \u306f\u6642\u9593 $t$ \u306e\u672a\u77e5\u95a2\u6570\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[19]:<\/div>\n
\n
\n
\n
# \u307e\u305a x \u3092\u95a2\u6570\u3068\u3057\u3066\u5ba3\u8a00<\/span>\r\nx<\/span> =<\/span> Function<\/span>(<\/span>'x'<\/span>)<\/span>\r\n\r\n# \u5b9a\u6570 omega0 > 0 \u3092\u5ba3\u8a00<\/span>\r\nvar<\/span>(<\/span>'omega0'<\/span>,<\/span> positive<\/span> =<\/span> True<\/span>)<\/span>\r\n\r\n# \u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5b9a\u7fa9<\/span>\r\neq<\/span> =<\/span> Eq<\/span>(<\/span>diff<\/span>(<\/span>x<\/span>(<\/span>t<\/span>),<\/span> t<\/span>,<\/span> 2<\/span>)<\/span> +<\/span> omega0<\/span>**<\/span>2<\/span> *<\/span> x<\/span>(<\/span>t<\/span>),<\/span> 0<\/span>)<\/span>\r\n# \u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u8868\u793a<\/span>\r\ndisplay<\/span>(<\/span>eq<\/span>)<\/span>\r\n\r\ndsolve<\/span>(<\/span>eq<\/span>,<\/span> x<\/span>(<\/span>t<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
$\\displaystyle \\omega_{0}^{2} x{\\left(t \\right)} + \\frac{d^{2}}{d t^{2}} x{\\left(t \\right)} = 0$<\/div>\n<\/div>\n
\n
Out[19]:<\/div>\n
$\\displaystyle x{\\left(t \\right)} = C_{1} \\sin{\\left(\\omega_{0} t \\right)} + C_{2} \\cos{\\left(\\omega_{0} t \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u521d\u671f\u6761\u4ef6\u306e\u8a2d\u5b9a\u4f8b<\/h5>\n
\u4e0a\u306e\u30bb\u30eb\u3067\uff0cdsolve()<\/code> \u3067\u89e3\u3044\u305f\u7d50\u679c\u306e $x(t)$ \u306b\u3042\u3089\u308f\u308c\u308b $C_1, C_2$ \u306f\u7a4d\u5206\u5b9a\u6570\u3067\u3059\u3002\uff082\u968e\u5fae\u5206\u306a\u306e\u3067\u7a4d\u5206\u5b9a\u6570\u306f2\u3064\u3002\uff09<\/p>\n
\u521d\u671f\u6761\u4ef6\u3092\u8ab2\u3059\u3053\u3068\u3067\u7a4d\u5206\u5b9a\u6570\u306e\u5024\u3092\u6c7a\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u540c\u3058\u554f\u984c\uff08$t$ \u3067\u306e\u5fae\u5206\u3092 $\\dot{}$\uff08\u30c9\u30c3\u30c8\uff09\u3067\u66f8\u3044\u3066\uff09<\/p>\n
$$ \\ddot{x} + \\omega_0^2 \\,x = 0$$<\/p>\n
\u3092\uff0c$t = 0$ \u3067 $\\displaystyle x(0) = x_0, \\dot{x}(0) = v_0$ \u3068\u3044\u3046\u521d\u671f\u6761\u4ef6\u3067\u89e3\u304d\u307e\u3059\u3002<\/p>\n
\u521d\u671f\u6761\u4ef6\u306f dsolve()<\/code> \u306e\u30aa\u30d7\u30b7\u30e7\u30f3\u3068\u3057\u3066 ics={}<\/code> \u306e\u3088\u3046\u306b\u8f9e\u66f8\u578b\u306e\u30c7\u30fc\u30bf\u5f62\u5f0f\u3067\u4e0e\u3048\u308b\u306e\u3067\u3057\u305f\u3002\u3053\u3053\u3067\u306f\uff0c<\/p>\n
$x(0) = x_0 \\Rightarrow$ x(0):x0<\/code>
\n$\\dot{x}(0) = v_0 \\Rightarrow$ x(t).diff(t).subs(t,0):v0<\/code><\/p>\n
\u3068\u3057\u3066\u3044\u307e\u3059\u3002\uff081\u968e\u5fae\u5206\u306b\u5bfe\u3059\u308b\u521d\u671f\u6761\u4ef6\u306e\u66f8\u304d\u65b9\u304c\u3059\u3050\u306b\u306f\u601d\u3044\u3064\u304d\u307e\u305b\u3093\u306d\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[20]:<\/div>\n
\n
\n
\n
# \uff082\u6587\u5b57\u4ee5\u4e0a\u306e\u5909\u6570\u306e\u5834\u5408\u306f\uff09\u5909\u6570\u3068\u3057\u3066\u5ba3\u8a00<\/span>\r\nvar<\/span>(<\/span>'x0 v0'<\/span>)<\/span>\r\n\r\ndsolve<\/span>(<\/span>eq<\/span>,<\/span> x<\/span>(<\/span>t<\/span>),<\/span> \r\n       ics<\/span> =<\/span> {<\/span>x<\/span>(<\/span>0<\/span>):<\/span>x0<\/span>,<\/span> \r\n              x<\/span>(<\/span>t<\/span>)<\/span>.<\/span>diff<\/span>(<\/span>t<\/span>)<\/span>.<\/span>subs<\/span>(<\/span>t<\/span>,<\/span>0<\/span>):<\/span>v0<\/span>}<\/span>\r\n      )<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[20]:<\/div>\n
$\\displaystyle x{\\left(t \\right)} = x_{0} \\cos{\\left(\\omega_{0} t \\right)} + \\frac{v_{0} \\sin{\\left(\\omega_{0} t \\right)}}{\\omega_{0}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
… \u3068\u3044\u3046\u308f\u3051\u3067\uff0c\u3053\u306e\u521d\u671f\u6761\u4ef6\u306b\u3088\u3063\u3066 $\\displaystyle C_2 = x_0, \\ C_1 = \\frac{v_0}{\\omega} $ \u3068\u6c42\u307e\u308a\u307e\u3057\u305f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u6700\u3082\u7c21\u5358\u306a\u5b9a\u6570\u4fc2\u65702\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h4>\n
\u3042\u3089\u305f\u3081\u3066 var('x')<\/code> \u3068\u3057\u3066\u72ec\u7acb\u5909\u6570\u3092 $x$\uff0cy = Function('y')<\/code> \u3068\u3057\u3066\u672a\u77e5\u95a2\u6570\u3092 $y(x)$ \u3068\u8868\u8a18\u3057\u3066\uff0c\u6700\u3082\u7c21\u5358\u306a\u5f62\u306e
\n2\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f<\/p>\n
$$y^{\\prime\\prime} + K\\, y = 0$$<\/p>\n
\u3092\u89e3\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n
$K$ \u3092var('K')<\/code> \u3067\u5358\u306b\u5909\u6570\u3068\u3057\u3066\u5ba3\u8a00\u3057\u305f\u5834\u5408\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[21]:<\/div>\n
\n
\n
\n
y<\/span> =<\/span> Function<\/span>(<\/span>'y'<\/span>)<\/span>\r\nvar<\/span>(<\/span>'x'<\/span>)<\/span>\r\nvar<\/span>(<\/span>'K'<\/span>)<\/span>\r\n\r\nsahen<\/span> =<\/span> diff<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 2<\/span>)<\/span> +<\/span> K<\/span>*<\/span>y<\/span>(<\/span>x<\/span>)<\/span>\r\ndsolve<\/span>(<\/span>sahen<\/span>,<\/span> y<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[21]:<\/div>\n
$\\displaystyle y{\\left(x \\right)} = C_{1} e^{- x \\sqrt{- K}} + C_{2} e^{x \\sqrt{- K}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u51fa\u529b\u7d50\u679c\u3092\u898b\u308b\u306b\uff0c\u5e73\u65b9\u6839\u306e\u4e2d\u304c\u6b63\u3067\u3042\u308b\u3053\u3068\u3092\u524d\u63d0\u306b\u3059\u308c\u3070 $K < 0$ \u3068\u4eee\u5b9a\u3057\u3066\u89e3\u3044\u3066\u3044\u308b\u3088\u3046\u3067\u3059\u3002\u305d\u3053\u3067\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b $K$ \u306e\u5024\u306b\u3064\u3044\u3066\u306f\u5834\u5408\u5206\u3051\u3092\u3057\u3066\u89e3\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n
$K > 0$ \u306e\u5834\u5408<\/h5>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[22]:<\/div>\n
\n
\n
\n
# K > 0 \u3068\u3059\u308b<\/span>\r\nvar<\/span>(<\/span>'K'<\/span>,<\/span> positive<\/span> =<\/span> True<\/span>)<\/span>\r\nsahen<\/span> =<\/span> diff<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 2<\/span>)<\/span> +<\/span> K<\/span>*<\/span>y<\/span>(<\/span>x<\/span>)<\/span>\r\n\r\ndsolve<\/span>(<\/span>sahen<\/span>,<\/span> y<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[22]:<\/div>\n
$\\displaystyle y{\\left(x \\right)} = C_{1} \\sin{\\left(\\sqrt{K} x \\right)} + C_{2} \\cos{\\left(\\sqrt{K} x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$K < 0$ \u306e\u5834\u5408<\/h5>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[23]:<\/div>\n
\n
\n
\n
# K < 0 \u3068\u3059\u308b<\/span>\r\nvar<\/span>(<\/span>'K'<\/span>,<\/span> negative<\/span> =<\/span> True<\/span>)<\/span>\r\nsahen<\/span> =<\/span> diff<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 2<\/span>)<\/span> +<\/span> K<\/span>*<\/span>y<\/span>(<\/span>x<\/span>)<\/span>\r\n\r\ndsolve<\/span>(<\/span>sahen<\/span>,<\/span> y<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[23]:<\/div>\n
$\\displaystyle y{\\left(x \\right)} = C_{1} e^{- x \\sqrt{- K}} + C_{2} e^{x \\sqrt{- K}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$K = 0$ \u306e\u5834\u5408<\/h5>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[24]:<\/div>\n
\n
\n
\n
var<\/span>(<\/span>'K'<\/span>)<\/span>\r\nsahen<\/span> =<\/span> diff<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 2<\/span>)<\/span> +<\/span> K<\/span>*<\/span>y<\/span>(<\/span>x<\/span>)<\/span>\r\n\r\ndsolve<\/span>(<\/span>sahen<\/span>.<\/span>subs<\/span>(<\/span>K<\/span>,<\/span>0<\/span>),<\/span> y<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[24]:<\/div>\n
$\\displaystyle y{\\left(x \\right)} = C_{1} + C_{2} x$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u25cb\u7df4\u7fd2\uff1a1\u3064\u306e\u89e3\u30673\u3064\u306e\u5834\u5408\u3092\u8868\u3059<\/h5>\n
$\\displaystyle y^{\\prime\\prime} + K\\, y = 0$ \u306e $K > 0$ \u306e\u89e3\u306f $\\displaystyle C_2 = A, \\ C_1 = \\frac{B}{\\sqrt{K}\\ }$ \u3068\u66f8\u304d\u76f4\u3059\u3068<\/p>\n
$$y = A \\cos \\left(\\sqrt{K}x \\right) + \\frac{B}{\\sqrt{K}\\ } \\sin \\left(\\sqrt{K}x \\right) $$<\/p>\n
\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002\u3053\u306e\u89e3\u3092\u4f7f\u3063\u3066\uff0c$K < 0$ \u306e\u5834\u5408\u3068 $K = 0$ \u306e\u5834\u5408\u306e\u89e3\u3092\u76f4\u63a5\u6c42\u3081\u3066\u307f\u3088\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[25]:<\/div>\n
\n
\n
\n
# \u5ff5\u306e\u305f\u3081\u306b<\/span>\r\nvar<\/span>(<\/span>'A B K x'<\/span>)<\/span>\r\n\r\ny<\/span> =<\/span> A<\/span>*<\/span>cos<\/span>(<\/span>sqrt<\/span>(<\/span>K<\/span>)<\/span>*<\/span>x<\/span>)<\/span> +<\/span> B<\/span>\/<\/span>sqrt<\/span>(<\/span>K<\/span>)<\/span>*<\/span>sin<\/span>(<\/span>sqrt<\/span>(<\/span>K<\/span>)<\/span>*<\/span>x<\/span>)<\/span>\r\ny<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[25]:<\/div>\n
$\\displaystyle A \\cos{\\left(\\sqrt{K} x \\right)} + \\frac{B \\sin{\\left(\\sqrt{K} x \\right)}}{\\sqrt{K}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u306e\u89e3\u3092\u305d\u306e\u307e\u307e\u4f7f\u3063\u3066 $K < 0$ \u306e\u5834\u5408\u3092\u8868\u793a\u3055\u308c\u308b\u306b\u306f\uff0c$K$ \u306e\u7d76\u5bfe\u5024 $|K| = $ abs(K)<\/code> \u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3053\u3068\u3092\u4f7f\u3046\u3002<\/p>\n
$$ \\sqrt{K} = \\sqrt{- |K|} = i \\sqrt{|K|}\\quad \\mbox{for} \\ \\ K < 0$$<\/p>\n
\u307e\u305f\uff0c$K=0$ \u306e\u5834\u5408\u306f\u6975\u9650 $\\displaystyle \\lim_{K \\rightarrow 0} y$ \u3092\u3068\u3063\u3066\u307f\u308c\u3070\u3088\u3044\u3067\u3042\u308d\u3046\u3002<\/p>\n
\u3068\u3044\u3046\u3053\u3068\u3067\uff0c$K > 0$ \u306e\u5834\u5408\u306e\u89e3\u30921\u3064\u899a\u3048\u3066\u304a\u3051\u3070\uff0c$K < 0$ \u306e\u5834\u5408\u3082 $K = 0$ \u306e\u5834\u5408\u3082\u305d\u308c\u304b\u3089\u5c0e\u304f\u3053\u3068\u304c\u3067\u304d\u3066\u5225\u9014\u899a\u3048\u3066\u304a\u304f\u5fc5\u8981\u306f\u306a\u3044\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[26]:<\/div>\n
\n
\n
\n
# \u4ee5\u4e0b\u306e\u3088\u3046\u306b .subs() \u3067\u4e00\u6642\u7684\u4ee3\u5165<\/span>\r\nsqrt<\/span>(<\/span>K<\/span>)<\/span>.<\/span>subs<\/span>(<\/span>K<\/span>,<\/span> -<\/span>abs<\/span>(<\/span>K<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[26]:<\/div>\n
$\\displaystyle i \\sqrt{\\left|{K}\\right|}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$K < 0$ \u306e\u5834\u5408\u306e $y$ \u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$K = 0$ \u306e\u5834\u5408\u306e $y$ \u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h4>\n
\u540c\u6b21\u65b9\u7a0b\u5f0f<\/h5>\n
\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
$$y^{\\prime\\prime} + 2 b\\, y^{\\prime} + c\\, y = 0$$<\/p>\n
$b, c$ \u306f\u5b9a\u6570\u3002<\/p>\n
SymPy \u3067\u3053\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u306b\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[27]:<\/div>\n
\n
\n
\n
y<\/span> =<\/span> Function<\/span>(<\/span>'y'<\/span>)<\/span>\r\nvar<\/span>(<\/span>'b c x'<\/span>)<\/span>\r\n\r\nsahen<\/span> =<\/span> diff<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 2<\/span>)<\/span> +<\/span> 2<\/span>*<\/span>b<\/span>*<\/span>diff<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span> +<\/span> c<\/span>*<\/span>y<\/span>(<\/span>x<\/span>)<\/span>\r\ndisplay<\/span>(<\/span>sahen<\/span>)<\/span>\r\n\r\ndsolve<\/span>(<\/span>sahen<\/span>,<\/span> y<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
$\\displaystyle 2 b \\frac{d}{d x} y{\\left(x \\right)} + c y{\\left(x \\right)} + \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)}$<\/div>\n<\/div>\n
\n
Out[27]:<\/div>\n
$\\displaystyle y{\\left(x \\right)} = C_{1} e^{x \\left(- b + \\sqrt{b^{2} – c}\\right)} + C_{2} e^{- x \\left(b + \\sqrt{b^{2} – c}\\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u4e0a\u306e\u7d50\u679c\u3092\u898b\u308b\u306b\uff0c$b^2 – c > 0$ \u3092\u6697\u9ed9\u306b\u4eee\u5b9a\u3057\u305f\u3068\u304d\u306e\u89e3\u306b\u306a\u3063\u3066\u3044\u308b\u3002\u305d\u308c\u4ee5\u5916\u306e\u5834\u5408\u306e\u89e3\u306f\u3069\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u306e\u3067\u3042\u308d\u3046\u304b…<\/p>\n
\u3068\u3053\u308d\u3067\uff0c\u3053\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u4e21\u8fba\u306b $e^{b x}$ \u3092\u304b\u3051\u308b\u3068<\/p>\n
\\begin{eqnarray}
\n0 &=& e^{b x} \\cdot \\left\\{ y^{\\prime\\prime} + 2 b\\, y^{\\prime} + c\\, y\\right\\} \\\\
\n&=& \\left( e^{b x} y \\right)^{\\prime\\prime} + \\left(c – b^2\\right) \\left( e^{b x} y \\right) \\\\
\n&\\equiv& Y^{\\prime\\prime} + K\\, Y
\n\\end{eqnarray}<\/p>\n
\u3068\u66f8\u304d\u63db\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u3053\u3067 $Y \\equiv e^{b x} y, \\ K \\equiv c – b^2$ \u3068\u3057\u305f\u3002<\/p>\n
\u3064\u307e\u308a<\/p>\n
$$Y^{\\prime\\prime} + K\\, Y = 0$$<\/p>\n
\u306e\u5f62\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u3051\u3070\u3088\u3044\u3053\u3068\u306b\u306a\u308a\uff0c\u3053\u308c\u306f\u3059\u3067\u306b $K = c -b^2 > 0$ \u306e\u5834\u5408\u306b\u306f\u4e00\u822c\u89e3\u304c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u3044\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n
\\begin{eqnarray}
\nY &=& A \\cos \\left(\\sqrt{K}x \\right) + \\frac{B}{\\sqrt{K}\\ } \\sin \\left(\\sqrt{K}x \\right) \\\\
\n\\therefore\\ \\ y = e^{-bx} Y &=& e^{-b x} \\left\\{A \\cos \\left(\\sqrt{K}x \\right) + \\frac{B}{\\sqrt{K}\\ } \\sin \\left(\\sqrt{K}x \\right) \\right\\}
\n\\end{eqnarray}<\/p>\n
\u305f\u3081\u3057\u306b\u3053\u308c\u3092\u95a2\u6570 y1(K)<\/code> \u3068\u3057\u3066\u5b9a\u7fa9\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[28]:<\/div>\n
\n
\n
\n
def<\/span> y1<\/span>(<\/span>K<\/span>):<\/span>\r\n    return<\/span> exp<\/span>(<\/span>-<\/span>b<\/span>*<\/span>x<\/span>)<\/span>*<\/span>(<\/span>A<\/span>*<\/span>cos<\/span>(<\/span>sqrt<\/span>(<\/span>K<\/span>)<\/span>*<\/span>x<\/span>)<\/span>+<\/span>B<\/span>\/<\/span>sqrt<\/span>(<\/span>K<\/span>)<\/span>*<\/span>sin<\/span>(<\/span>sqrt<\/span>(<\/span>K<\/span>)<\/span>*<\/span>x<\/span>))<\/span>\r\n\r\ndisplay<\/span>(<\/span>y1<\/span>(<\/span>K<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
$\\displaystyle \\left(A \\cos{\\left(\\sqrt{K} x \\right)} + \\frac{B \\sin{\\left(\\sqrt{K} x \\right)}}{\\sqrt{K}}\\right) e^{- b x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u25cb\u7df4\u7fd2\uff1a\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3<\/h5>\n
\u540c\u6b21\u65b9\u7a0b\u5f0f<\/p>\n
$$ y^{\\prime\\prime} + 2 b\\, y^{\\prime} + c\\, y = 0$$<\/p>\n
\u306e\u4e00\u822c\u89e3\u3092\uff0c<\/p>\n
    \n
  1. $c – b^2 > 0$,<\/li>\n
  2. $c – b^2 < 0$,<\/li>\n
  3. $c – b^2 = 0$<\/li>\n<\/ol>\n
    \u305d\u308c\u305e\u308c\u306e\u5834\u5408\u306b y1(K)<\/code> \u3092\u4f7f\u3063\u3066\u8868\u305b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
      \n
    1. $c – b^2 > 0$ \u306e\u5834\u5408<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[29]:<\/div>\n
      \n
      \n
      \n
      y1<\/span>(<\/span>c<\/span>-<\/span>b<\/span>**<\/span>2<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[29]:<\/div>\n
      $\\displaystyle \\left(A \\cos{\\left(x \\sqrt{- b^{2} + c} \\right)} + \\frac{B \\sin{\\left(x \\sqrt{- b^{2} + c} \\right)}}{\\sqrt{- b^{2} + c}}\\right) e^{- b x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
        \n
      1. $c – b^2 < 0$ \u306e\u5834\u5408<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        \n
        In\u00a0[\u00a0]:<\/div>\n
        \n
        \n
        \n
         \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
        \n
        <\/div>\n
        \n
        \n
          \n
        1. $c – b^2 = 0$ \u306e\u5834\u5408<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[\u00a0]:<\/div>\n
          \n
          \n
          \n
           \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u975e\u540c\u6b21\u65b9\u7a0b\u5f0f<\/h5>\n
          \u975e\u540c\u6b21\u9805 $R(x)$ \u304c\u4e00\u822c\u306b\u30bc\u30ed\u3067\u306a\u3044\u3068\u304d\u306b\u306f<\/p>\n
          $$y^{\\prime\\prime} + 2 b\\, y^{\\prime} + c\\, y = R(x)$$<\/p>\n
          \u306e\u5f62\u306b\u306a\u308a\uff0c\u3053\u308c\u3092\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u300c\u975e\u540c\u6b21\u300d\u65b9\u7a0b\u5f0f<\/strong><\/span>\u3068\u547c\u3076\u3002<\/p>\n
          \u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3\u3092\u4eba\u529b\u3067\u6c42\u3081\u308b\u306b\u306f\uff0c\u3042\u305d\u3053<\/a>\u3067\u307e\u3068\u3081\u3066\u3044\u308b\u3088\u3046\u306b\u7d50\u69cb\u624b\u9593\u304c\u304b\u304b\u308b\u306e\u3060\u304c\uff0cSymPy \u3067\u306f\u540c\u6b21\u65b9\u7a0b\u5f0f\u3082\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u3082\uff0c\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u969b\u306b\u306f dsolve()<\/code> \u3092\u4f7f\u3046\u3060\u3051\u3067\u3042\u308a\uff0c\u4eba\u9593\u306e\u624b\u9593\u3068\u3057\u3066\u306f\u540c\u3058\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4f8b\uff1a\u5f37\u5236\u632f\u52d5<\/h5>\n
          \u56fa\u6709\u632f\u52d5\u6570 $\\omega_0$ \u306e\u8abf\u548c\u632f\u52d5\u5b50\u306b\uff0c\u632f\u52d5\u6570 $\\omega$ \u306e\u5468\u671f\u7684\u306a\u5916\u529b $F(\\omega t)$ \u304c\u50cd\u304f\u5834\u5408\uff0c\u3053\u306e\u904b\u52d5\u306f\u5f37\u5236\u632f\u52d5\u3068\u547c\u3070\u308c\uff0c\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5f62\u306b\u306a\u308b\u3002<\/p>\n
          $$\\ddot{x} + \\omega_0^2 \\,x = \\frac{F(\\omega t)}{m}$$<\/p>\n
          \u7c21\u5358\u306e\u305f\u3081\u306b\uff0c$\\displaystyle \\frac{F(\\omega t)}{m} = \\sin \\omega t$ \u3068\u3059\u308b\u3068\uff0c
          \n\u5f37\u5236\u632f\u52d5\u306e\u904b\u52d5\u65b9\u7a0b\u5f0f\u306f<\/p>\n
          $$\\ddot{x} + \\omega_0^2 \\,x = \\sin \\omega t$$<\/p>\n
          \u3053\u308c\u306f\u975e\u540c\u6b21\u9805 $\\sin \\omega t$ \u3092\u3082\u3064\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u3067\u3042\u308b\u3002<\/p>\n
          \u3053\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f SymPy \u3067\u306f dsolve()<\/code> \u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u89e3\u3051\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[30]:<\/div>\n
          \n
          \n
          \n
          x<\/span> =<\/span> Function<\/span>(<\/span>'x'<\/span>)<\/span>\r\nvar<\/span>(<\/span>'omega0 omega'<\/span>,<\/span> positive<\/span> =<\/span> True<\/span>)<\/span>\r\n\r\neq<\/span> =<\/span> Eq<\/span>(<\/span>diff<\/span>(<\/span>x<\/span>(<\/span>t<\/span>),<\/span> t<\/span>,<\/span> 2<\/span>)<\/span>+<\/span>omega0<\/span>**<\/span>2<\/span> *<\/span> x<\/span>(<\/span>t<\/span>),<\/span> sin<\/span>(<\/span>omega<\/span>*<\/span>t<\/span>))<\/span>\r\neq<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[30]:<\/div>\n
          $\\displaystyle \\omega_{0}^{2} x{\\left(t \\right)} + \\frac{d^{2}}{d t^{2}} x{\\left(t \\right)} = \\sin{\\left(\\omega t \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[31]:<\/div>\n
          \n
          \n
          \n
          dsolve<\/span>(<\/span>eq<\/span>,<\/span> x<\/span>(<\/span>t<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[31]:<\/div>\n
          $\\displaystyle x{\\left(t \\right)} = C_{1} \\sin{\\left(\\omega_{0} t \\right)} + C_{2} \\cos{\\left(\\omega_{0} t \\right)} – \\frac{\\sin{\\left(\\omega t \\right)}}{\\omega^{2} – \\omega_{0}^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4f8b\uff1a\u5171\u9cf4<\/h5>\n
          \u4e0a\u306e\u51fa\u529b\u7d50\u679c\u304b\u3089\uff0c$\\omega = \\omega_0$ \u306e\u5834\u5408\u306f\u5206\u6bcd\u304c\u30bc\u30ed\u306b\u306a\u308b\u305f\u3081\uff0c\u5225\u9014\u8a08\u7b97\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\uff0c
          \neq.subs(omega, omega0)<\/code> \u3067\u65b9\u7a0b\u5f0f eq<\/code> \u306e\u4e2d\u306e omega<\/code> \u3092 omega0<\/code> \u306b\u5165\u308c\u66ff\u3048\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[32]:<\/div>\n
          \n
          \n
          \n
          display<\/span>(<\/span>eq<\/span>.<\/span>subs<\/span>(<\/span>omega<\/span>,<\/span> omega0<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          <\/div>\n
          $\\displaystyle \\omega_{0}^{2} x{\\left(t \\right)} + \\frac{d^{2}}{d t^{2}} x{\\left(t \\right)} = \\sin{\\left(\\omega_{0} t \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u3053\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092 dsolve()<\/code> \u3067\u89e3\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[33]:<\/div>\n
          \n
          \n
          \n
          dsolve<\/span>(<\/span>eq<\/span>.<\/span>subs<\/span>(<\/span>omega<\/span>,<\/span> omega0<\/span>),<\/span> x<\/span>(<\/span>t<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[33]:<\/div>\n
          $\\displaystyle x{\\left(t \\right)} = C_{2} \\sin{\\left(\\omega_{0} t \\right)} + \\left(C_{1} – \\frac{t}{2 \\omega_{0}}\\right) \\cos{\\left(\\omega_{0} t \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u7a4d\u5206\u5b9a\u6570 $C_1, C_2$ \u304c\u3064\u304f\u306e\u306f\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u57fa\u672c\u89e3\u3002\u305d\u308c\u4ee5\u5916\u306b\u6642\u9593 $t$ \u306b\u6bd4\u4f8b\u3057\u3066\u5358\u8abf\u5897\u52a0\u3059\u308b\u632f\u5e45\u3092\u6301\u3064\u9805\u304c\u73fe\u308c\u308b\u3002<\/p>\n
          \u8abf\u548c\u632f\u52d5\u5b50\u306e\u56fa\u6709\u632f\u52d5\u6570\u3068\u540c\u3058\u30bf\u30a4\u30df\u30f3\u30b0\u306e\u5468\u671f\u7684\u5916\u529b\u3092\u52a0\u3048\u305f\u5f37\u5236\u632f\u52d5\u306b\u73fe\u308c\u308b\u3053\u306e\u73fe\u8c61\u306f\u5171\u9cf4<\/strong>\u3068\u547c\u3070\u308c\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u25cb\u7df4\u7fd2\uff1a\u6975\u9650\u3068\u3057\u3066\u306e\u5171\u9cf4<\/h5>\n
          \u5f37\u5236\u632f\u52d5<\/p>\n
          $$\\ddot{x} + \\omega_0^2 \\,x = \\sin \\omega t$$<\/p>\n
          \u306e\u89e3\u306f<\/p>\n
          $$x{\\left(t \\right)} = C_{1} \\sin{\\left(\\omega_{0} t \\right)} + C_{2} \\cos{\\left(\\omega_{0} t \\right)} + \\frac{\\sin{\\left(\\omega t \\right)}}{ \\omega_{0}^{2}- \\omega^{2} }$$<\/p>\n
          \u3067\u3042\u308b\u306e\u3067\uff0c$\\omega \\rightarrow \\omega_0$ \u306e\u5834\u5408\u306f\u5206\u6bcd\u304c\u30bc\u30ed\u306b\u306a\u3063\u3066\u5927\u5909\u306a\u3053\u3068\u306b\u306a\u308b\u3068\u601d\u3044\u304d\u3084\uff0c<\/p>\n
          $$\\ddot{x} + \\omega_0^2 \\,x = \\sin \\omega_0 t$$<\/p>\n
          \u306e\u89e3\u306f\uff0c\u4f55\u4e8b\u3082\u306a\u304b\u3063\u305f\u304b\u306e\u3088\u3046\u306b<\/p>\n
          $$x{\\left(t \\right)} = C_{2} \\sin{\\left(\\omega_{0} t \\right)} + \\left(C_{1} -\\frac{t}{2 \\omega_{0}}\\right) \\cos{\\left(\\omega_{0} t \\right)}$$<\/p>\n
          \u3068\u306a\u308b\u3002\u3053\u308c\u3092 $\\omega \\rightarrow \\omega_0$ \u306e\u6975\u9650\u3068\u3057\u3066\u7406\u89e3\u3057\u305f\u3044\u3068\u304d\u306b\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u305f\u3089\u3069\u3046\u304b\uff0c\u3068\u3044\u3046\u8a71\u3002<\/p>\n
          \u307e\u305a\uff0c$\\omega \\neq \\omega_0$ \u306e\u5834\u5408\u306e\u89e3\u3092\uff0c\u7a4d\u5206\u5b9a\u6570 $C_1$ \u3092\u3061\u3087\u3063\u3068\u5909\u3048\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3059\u308b\u3002<\/p>\n
          \\begin{eqnarray}
          \nx{\\left(t \\right)} &=& C_{1} \\sin{\\left(\\omega_{0} t \\right)} + C_{2} \\cos{\\left(\\omega_{0} t \\right)} + \\frac{\\sin{\\left(\\omega t \\right)}}{ \\omega_{0}^{2} -\\omega^{2} } \\\\
          \n&=& \\left(C^{\\prime}_1 -\\frac{1}{\\omega_0^2 -\\omega^2}\\right) \\sin{\\left(\\omega_{0} t \\right)}
          \n+ C_{2} \\cos{\\left(\\omega_{0} t \\right)}
          \n+ \\frac{\\sin{\\left(\\omega t \\right)}}{ \\omega_{0}^{2}- \\omega^{2} }\\\\
          \n&=& C^{\\prime}_1 \\sin{\\left(\\omega_{0} t \\right)} + C_{2} \\cos{\\left(\\omega_{0} t \\right)}
          \n+ \\frac{\\sin{\\left(\\omega t \\right)} -\\sin{\\left(\\omega_{0} t \\right)}}{ \\omega_{0}^{2} -\\omega^{2} }
          \n\\end{eqnarray}<\/p>\n
          \u3042\u3089\u305f\u3081\u3066\u7a4d\u5206\u5b9a\u6570\u3092 $C_1^{\\prime} \\rightarrow C_1$ \u3092\u7f6e\u304d\u76f4\u3057\u3066\u66f8\u304d\u76f4\u3059\u3068\uff0c$\\omega \\neq \\omega_0$ \u306e\u5834\u5408\u306e\u5f37\u5236\u632f\u52d5\u306e\u89e3\u306f\uff0c<\/p>\n
          \\begin{eqnarray}
          \nx{\\left(t \\right)} &=&
          \nC_1 \\sin{\\left(\\omega_{0} t \\right)} + C_{2} \\cos{\\left(\\omega_{0} t \\right)}
          \n+ \\frac{\\sin{\\left(\\omega t \\right)} -\\sin{\\left(\\omega_{0} t \\right)}}{ \\omega_{0}^{2} -\\omega^{2} }
          \n\\end{eqnarray}<\/p>\n
          \u3068\u66f8\u3044\u3066\u3082\u3088\u3044\u3053\u3068\u306b\u306a\u308b\u3002\uff08\u7a4d\u5206\u5b9a\u6570\u306f\u4efb\u610f\u3067\u3042\u3063\u305f\u304b\u3089\uff09<\/p>\n
          \u3053\u306e\u89e3\u306e\u6975\u9650 $\\omega \\rightarrow \\omega_0$ \u306e\u6975\u9650\u3092\u3068\u308b\u3068\uff0c\u5171\u9cf4\u306e\u89e3\u304c\u5f97\u3089\u308c\u308b\u3053\u3068\u3092\u793a\u305b\u3002<\/p>\n
          $$\\lim_{\\omega \\rightarrow \\omega_0} \\frac{\\sin{\\left(\\omega t \\right)} -\\sin{\\left(\\omega_{0} t \\right)}}{ \\omega_{0}^{2}- \\omega^{2} } = \\cdots $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[\u00a0]:<\/div>\n
          \n
          \n
          \n
          <\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u4eba\u529b\u3067\u3053\u306e\u6975\u9650\u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u306f\uff0c<\/p>\n
          $$\\sin A -\\sin B = 2 \\cos \\frac{A+B}{2} \\cdot \\sin\\frac{A -B}{2}$$<\/p>\n
          \u3092\u4f7f\u3063\u3066\uff0c<\/p>\n
          \\begin{eqnarray}
          \n\\frac{\\sin{\\left(\\omega t \\right)} -\\sin{\\left(\\omega_{0} t \\right)}}{ \\omega_{0}^{2} -\\omega^{2} } &=&
          \n\\frac{2}{\\omega_{0}^{2} -\\omega^{2} }
          \n\\cos \\frac{(\\omega+\\omega_0) t}{2} \\cdot
          \n\\sin \\frac{(\\omega -\\omega_0) t}{2}\\\\
          \n&=& -\\frac{1}{\\omega+\\omega_0} \\cos \\frac{(\\omega+\\omega_0) t}{2} \\cdot
          \n\\frac{2}{\\omega -\\omega_0} \\sin \\frac{(\\omega -\\omega_0) t}{2}
          \n\\end{eqnarray}<\/p>\n
          \u3053\u306e\u5f62\u306b\u3059\u308c\u3070\uff0c$\\omega \\rightarrow \\omega_0$ \u306e\u6975\u9650\u3067<\/p>\n
          \\begin{eqnarray}
          \n-\\frac{1}{2 \\omega_0} \\cos \\omega_0 t \\cdot
          \n\\lim_{\\omega \\rightarrow \\omega_0} \\frac{2}{\\omega -\\omega_0} \\sin \\frac{(\\omega -\\omega_0) t}{2} &=& -\\frac{t}{2 \\omega_0}\\cos \\omega_0 t
          \n\\end{eqnarray}<\/p>\n
          \u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u3084\u3059\u3044\u3067\u3042\u308d\u3046\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u25cb\u7df4\u7fd2\uff1a\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3<\/h5>\n
          \u975e\u540c\u6b21\u65b9\u7a0b\u5f0f<\/p>\n
          $$y^{\\prime\\prime} + 2 b\\, y^{\\prime} + y = R(x) \\equiv \\sin x, \\quad (b > 0)$$<\/p>\n
          \u306e\u4e00\u822c\u89e3\u306f\uff0c\u53f3\u8fba\u306e\u975e\u540c\u6b21\u9805\u3092 $R(x)=0$ \u3068\u3057\u305f\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\uff08\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u57fa\u672c\u89e3\u306e\u7dda\u5f62\u548c\uff09\u306b\uff0c\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3\u3092\u52a0\u3048\u305f\u3082\u306e\u3067\u3059\u3002<\/p>\n
          \u307e\u305a\u306f SymPy \u306e dsolve()<\/code> \u3067\u89e3\u304b\u305b\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[34]:<\/div>\n
          \n
          \n
          \n
          y<\/span> =<\/span> Function<\/span>(<\/span>'y'<\/span>)<\/span>\r\nvar<\/span>(<\/span>'b c x'<\/span>)<\/span>\r\n\r\neq<\/span> =<\/span> Eq<\/span>(<\/span>diff<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 2<\/span>)<\/span>\r\n        +<\/span> 2<\/span>*<\/span>b<\/span>*<\/span>diff<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>\r\n        +<\/span> y<\/span>(<\/span>x<\/span>),<\/span> sin<\/span>(<\/span>x<\/span>))<\/span>\r\ndisplay<\/span>(<\/span>eq<\/span>)<\/span>\r\n\r\nsol<\/span> =<\/span> dsolve<\/span>(<\/span>eq<\/span>,<\/span> y<\/span>(<\/span>x<\/span>))<\/span>\r\nsol<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          <\/div>\n
          $\\displaystyle 2 b \\frac{d}{d x} y{\\left(x \\right)} + y{\\left(x \\right)} + \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)} = \\sin{\\left(x \\right)}$<\/div>\n<\/div>\n
          \n
          Out[34]:<\/div>\n
          $\\displaystyle y{\\left(x \\right)} = C_{1} e^{x \\left(- b + \\sqrt{b^{2} – 1}\\right)} + C_{2} e^{- x \\left(b + \\sqrt{b^{2} – 1}\\right)} – \\frac{\\cos{\\left(x \\right)}}{2 b}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u4e0a\u306e\u51fa\u529b\u7d50\u679c\u3092\u307f\u308b\u3068\uff0c$b^2 – 1 > 0$ \u3092\u6697\u9ed9\u306e\u3046\u3061\u306b\u4eee\u5b9a\u3057\u3066\u89e3\u3044\u3066\u3044\u308b\u3088\u3046\u3067\u3059\u3002<\/p>\n
          \u4ee5\u4e0b\u306e\u3088\u3046\u306b\uff0c\u3053\u306e\u89e3 sol.rhs<\/code> \u3092\u8fd4\u3059\u95a2\u6570 y2(K)<\/code> \u3092\u5b9a\u7fa9\u3057\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[35]:<\/div>\n
          \n
          \n
          \n
          def<\/span> y2<\/span>(<\/span>K<\/span>):<\/span>\r\n    return<\/span> sol<\/span>.<\/span>rhs<\/span>.<\/span>subs<\/span>(<\/span>b<\/span>**<\/span>2<\/span>-<\/span>1<\/span>,<\/span> K<\/span>)<\/span>\r\ny2<\/span>(<\/span>K<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[35]:<\/div>\n
          $\\displaystyle C_{1} e^{x \\left(\\sqrt{K} – b\\right)} + C_{2} e^{- x \\left(\\sqrt{K} + b\\right)} – \\frac{\\cos{\\left(x \\right)}}{2 b}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          $b^2-1>0$ \u306e\u5834\u5408\u306b\u306f K<\/code> \u306b b**2-1<\/code> \u3092\u5165\u308c\u3066 y2(b**2-1)<\/code> \u3068\u3057\u3066\u3084\u308b\u3068\u3088\u3055\u305d\u3046\u3067\u3059\u304c…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[36]:<\/div>\n
          \n
          \n
          \n
          y2<\/span>(<\/span>b<\/span>**<\/span>2<\/span>-<\/span>1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[36]:<\/div>\n
          $\\displaystyle C_{1} e^{x \\left(- b + \\sqrt{b^{2} – 1}\\right)} + C_{2} e^{- x \\left(b + \\sqrt{b^{2} – 1}\\right)} – \\frac{\\cos{\\left(x \\right)}}{2 b}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          $b^2-1<0$ \u306e\u5834\u5408\u306b\u306f y2(-abs(1-b**2))<\/code> \u3068\u3057\u3066\u3084\u3063\u3066\u3082\uff0c\u865a\u6570\u5358\u4f4d $i$ \u3092\u542b\u3093\u3060\u6307\u6570\u95a2\u6570\u306e\u307e\u307e\u3067\u3059\u3057…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[37]:<\/div>\n
          \n
          \n
          \n
          y2<\/span>(<\/span>-<\/span>abs<\/span>(<\/span>1<\/span>-<\/span>b<\/span>**<\/span>2<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[37]:<\/div>\n
          $\\displaystyle C_{1} e^{x \\left(- b + i \\sqrt{\\left|{b^{2} – 1}\\right|}\\right)} + C_{2} e^{- x \\left(b + i \\sqrt{\\left|{b^{2} – 1}\\right|}\\right)} – \\frac{\\cos{\\left(x \\right)}}{2 b}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          $b^2-1=0$ \u306e\u5834\u5408\u3068\u3057\u3066 y2(0).subs(b, 1)<\/code> \u3092\u8868\u793a\u3055\u305b\u308b\u3068\uff0c\u9593\u9055\u3063\u305f\u89e3\u304c\u51fa\u3066\u304f\u308b\u59cb\u672b\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[38]:<\/div>\n
          \n
          \n
          \n
          y2<\/span>(<\/span>0<\/span>)<\/span>.<\/span>subs<\/span>(<\/span>b<\/span>,<\/span> 1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[38]:<\/div>\n
          $\\displaystyle C_{1} e^{- x} + C_{2} e^{- x} – \\frac{\\cos{\\left(x \\right)}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u4e0a\u8a18\u306e\u89e3\u304c\u9593\u9055\u3063\u3066\u3044\u308b\u3053\u3068\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\uff0c\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u6bb5\u968e\u3067
          \neq.subs(b, 1)<\/code> \u3068\u3057\u3066\u304b\u3089\u89e3\u3044\u305f\u89e3\u3092\u898b\u308c\u3070\u660e\u3089\u304b\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[39]:<\/div>\n
          \n
          \n
          \n
          dsolve<\/span>(<\/span>eq<\/span>.<\/span>subs<\/span>(<\/span>b<\/span>,<\/span> 1<\/span>),<\/span> y<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[39]:<\/div>\n
          $\\displaystyle y{\\left(x \\right)} = \\left(C_{1} + C_{2} x\\right) e^{- x} – \\frac{\\cos{\\left(x \\right)}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u3053\u3053\u306f\u3084\u306f\u308a\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u4e00\u65e6\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u5909\u5f62\u3057\u3066\u3084\u308a\u307e\u3059\u3002<\/p>\n
          \\begin{eqnarray}
          \ny^{\\prime\\prime} + 2 b\\, y^{\\prime} + y &=& \\sin x \\\\
          \ne^{b x} \\cdot \\left\\{y^{\\prime\\prime} + 2 b\\, y^{\\prime} + y \\right\\} &=& e^{b x}\\sin x \\\\
          \n\\left(e^{b x} y\\right)^{\\prime\\prime} + \\left(1 -b^2\\right) \\left(e^{b x} y\\right)&=& e^{b x}\\sin x \\\\
          \n\\therefore\\ \\ Y^{\\prime\\prime} + K\\, Y &=& e^{b x}\\sin x
          \n\\end{eqnarray}<\/p>\n
          \u3053\u3053\u3067\uff0c$\\displaystyle Y \\equiv e^{b x} y, \\ K \\equiv 1 -b^2$ \u3068\u3057\u305f\u3002<\/p>\n
          \u3053\u308c\u3092\u307e\u305a\u4ee5\u4e0b\u306e\u3088\u3046\u306b $Y$ \u306b\u3064\u3044\u3066\u89e3\u304f\u3002<\/p>\n
          \u307e\u305a\u306f $0<b<1$ \u3059\u306a\u308f\u3061 $K = 1 -b^2 >0$ \u306e\u5834\u5408\uff1a<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[40]:<\/div>\n
          \n
          \n
          \n
          Y<\/span> =<\/span> Function<\/span>(<\/span>'Y'<\/span>)<\/span>\r\nvar<\/span>(<\/span>'K'<\/span>,<\/span> positive<\/span>=<\/span>True<\/span>)<\/span>\r\n\r\neq<\/span> =<\/span> Eq<\/span>(<\/span>diff<\/span>(<\/span>Y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 2<\/span>)<\/span>+<\/span> K<\/span>*<\/span>Y<\/span>(<\/span>x<\/span>),<\/span> exp<\/span>(<\/span>b<\/span>*<\/span>x<\/span>)<\/span>*<\/span>sin<\/span>(<\/span>x<\/span>))<\/span>\r\n\r\nsol1<\/span> =<\/span> dsolve<\/span>(<\/span>eq<\/span>,<\/span> Y<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          $Y$ \u304c\u6c42\u307e\u3063\u305f\u3089\uff0c$K=1-b^2$ \u3067\u3042\u308b\u304b\u3089 .subs(K, 1-b**2)<\/code> \u3067 $b$ \u3092\u4f7f\u3063\u305f\u8868\u793a\u306b\u623b\u3057\uff0c\u7c21\u5358\u5316 .simplify()<\/code>…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[41]:<\/div>\n
          \n
          \n
          \n
          sol1<\/span>.<\/span>rhs<\/span>.<\/span>subs<\/span>(<\/span>K<\/span>,<\/span>1<\/span>-<\/span>b<\/span>**<\/span>2<\/span>)<\/span>.<\/span>simplify<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[41]:<\/div>\n
          $\\displaystyle C_{1} \\sin{\\left(x \\sqrt{1 – b^{2}} \\right)} + C_{2} \\cos{\\left(x \\sqrt{1 – b^{2}} \\right)} – \\frac{e^{b x} \\cos{\\left(x \\right)}}{2 b}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          \u6700\u5f8c\u306b $y = e^{-bx} Y$ \u3067\u3042\u3063\u305f\u304b\u3089 exp(-b*x)<\/code> \u3092\u304b\u3051\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[42]:<\/div>\n
          \n
          \n
          \n
          Eq<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> (<\/span>exp<\/span>(<\/span>-<\/span>b<\/span>*<\/span>x<\/span>)<\/span>*<\/span>sol1<\/span>.<\/span>rhs<\/span>.<\/span>subs<\/span>(<\/span>K<\/span>,<\/span>1<\/span>-<\/span>b<\/span>**<\/span>2<\/span>)<\/span>.<\/span>simplify<\/span>())<\/span>.<\/span>expand<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[42]:<\/div>\n
          $\\displaystyle y{\\left(x \\right)} = C_{1} e^{- b x} \\sin{\\left(x \\sqrt{1 – b^{2}} \\right)} + C_{2} e^{- b x} \\cos{\\left(x \\sqrt{1 – b^{2}} \\right)} – \\frac{\\cos{\\left(x \\right)}}{2 b}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          $b>1$ \u3059\u306a\u308f\u3061 $K = 1-b^2 < 0$ \u306e\u5834\u5408\u306b\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[43]:<\/div>\n
          \n
          \n
          \n
          var<\/span>(<\/span>'K'<\/span>,<\/span> negative<\/span>=<\/span>True<\/span>)<\/span>\r\n\r\neq<\/span> =<\/span> Eq<\/span>(<\/span>diff<\/span>(<\/span>Y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 2<\/span>)<\/span>+<\/span> K<\/span>*<\/span>Y<\/span>(<\/span>x<\/span>),<\/span> exp<\/span>(<\/span>b<\/span>*<\/span>x<\/span>)<\/span>*<\/span>sin<\/span>(<\/span>x<\/span>))<\/span>\r\n\r\nsol2<\/span> =<\/span> dsolve<\/span>(<\/span>eq<\/span>,<\/span> Y<\/span>(<\/span>x<\/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
          Eq<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> (<\/span>exp<\/span>(<\/span>-<\/span>b<\/span>*<\/span>x<\/span>)<\/span>*<\/span>sol2<\/span>.<\/span>rhs<\/span>.<\/span>subs<\/span>(<\/span>K<\/span>,<\/span>1<\/span>-<\/span>b<\/span>**<\/span>2<\/span>)<\/span>.<\/span>simplify<\/span>())<\/span>.<\/span>expand<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[44]:<\/div>\n
          $\\displaystyle y{\\left(x \\right)} = C_{1} e^{- b x} e^{- x \\sqrt{b^{2} – 1}} + C_{2} e^{- b x} e^{x \\sqrt{b^{2} – 1}} – \\frac{\\cos{\\left(x \\right)}}{2 b}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          <\/div>\n
          \n
          \n
          $b = 1$ \u3059\u306a\u308f\u3061 $K = 1-b^2 = 0$ \u306e\u5834\u5408\u306b\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[45]:<\/div>\n
          \n
          \n
          \n
          sol3<\/span> =<\/span> dsolve<\/span>(<\/span>eq<\/span>.<\/span>subs<\/span>(<\/span>K<\/span>,<\/span>0<\/span>)<\/span>.<\/span>subs<\/span>(<\/span>b<\/span>,<\/span>1<\/span>),<\/span> Y<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          In\u00a0[46]:<\/div>\n
          \n
          \n
          \n
          Eq<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> (<\/span>exp<\/span>(<\/span>-<\/span>x<\/span>)<\/span>*<\/span>sol3<\/span>.<\/span>rhs<\/span>)<\/span>.<\/span>expand<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
          \n
          \n
          \n
          Out[46]:<\/div>\n
          $\\displaystyle y{\\left(x \\right)} = C_{1} e^{- x} + C_{2} x e^{- x} – \\frac{\\cos{\\left(x \\right)}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
          SymPy \u30671\u968e\u7dda\u5f62\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u30842\u968e\u5b9a\u6570\u4fc2\u6570\u7dda\u5f62\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u6790\u7684\u306b\u89e3\u304f\u3002<\/p>
          \u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":9502,"menu_order":40,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-9659","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\/9659","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=9659"}],"version-history":[{"count":8,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/9659\/revisions"}],"predecessor-version":[{"id":9677,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/9659\/revisions\/9677"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/9502"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=9659"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}