<\/p>\n
\u4ee5\u4e0b\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3002$$ \\frac{dy}{dx} + \\frac{y}{x} = \\frac{\\sin x}{x} $$<\/p>\n
\u5de6\u8fba\u306b \\(g(x)\\) \u3092\u304b\u3051\u3066<\/p>\n
$$ g(x) \\left( \\frac{dy}{dx} + \\frac{y}{x}\\right) = \\frac{d}{dx}\\bigl(g(x)\\,y\\bigr)$$<\/p>\n
\u3068\u306a\u308b\u3088\u3046\u306b\u3059\u308b\u306b\u306f\uff0c<\/p>\n
$$ \\frac{dg}{dx} = \\frac{1}{x} g(x) $$<\/p>\n
\u3092\u6e80\u305f\u305b\u3070\u3088\u3044\u3002\u3053\u308c\u304b\u3089 \\(g(x) = x\\) \u3068\u3059\u308c\u3070\u3088\u3044\u3053\u3068\u304c\u308f\u304b\u308b\u3088\u306d\u3002\u4e01\u5be7\u306b\u3084\u3063\u3066\u307f\u308b\u3068\uff0c\u5909\u6570\u5206\u96e2\u6cd5\u3067<\/p>\n
$$\\frac{dg}{g} = \\frac{dx}{x}$$<\/p>\n
<\/p>\n
$$\\int \\frac{dg}{g} = \\int \\frac{dx}{x}$$<\/p>\n
\u53b3\u5bc6\u306b\u306f<\/p>\n
$$ \\ln |g| = \\ln |x| $$<\/p>\n
\u306a\u3093\u3060\u308d\u3046\u3051\u3069\uff0c\u3069\u3061\u3089\u3082\u6b63\u3067\u3042\u308b\u3068\u3057\u3066<\/p>\n
$$ \\ln g = \\ln x $$<\/p>\n
\u3064\u307e\u308a\uff0c\\( g(x) = x \\) \u3002<\/p>\n
<\/p>\n
<\/p>\n
\u3067\uff0c\u5143\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u7a4d\u5206\u56e0\u5b50 \\(g(x) = x\\) \u3092\u304b\u3051\u308b\u3068\uff0c<\/p>\n
$$ \\frac{d}{dx}\\bigl(x\\, y(x)\\bigr) = \\sin x $$<\/p>\n
\u3053\u306e\u4e21\u8fba\u3092\u7a4d\u5206\u3057\u3066\uff0c<\/p>\n
$$ x\\, y(x) = \\int \\sin x \\,dx = -\\cos x + C$$<\/p>\n
$$\\therefore y(x) = \\frac{-\\cos x + C}{x} $$<\/p>\n
$$ \\frac{dy}{dx} + \\frac{y}{x} = \\frac{\\sin x}{x} $$<\/p>\n
\u3088\u308a\uff0c\u7a4d\u5206\u56e0\u5b50\u6cd5\u306e\u578b\u306b\u3042\u3066\u306f\u3081\u3066\u6a5f\u68b0\u7684\u306b\u89e3\u304f\u306b\u306f\uff0c<\/p>\n
$$ P(x) = \\frac{1}{x}, \\qquad Q(x) = \\frac{\\sin x}{x} $$<\/p>\n
\u3068\u3057\u3066\uff0c\u7a4d\u5206\u56e0\u5b50 \\(g(x) \\) \u306f<\/p>\n
$$ g(x) = \\exp\\left\\{ \\int P(x) dx \\right\\} = \\exp\\left\\{ \\ln |x| \\right\\} = |x|$$<\/p>\n
\\( g(x) = x\\) \u307e\u305f\u306f \\( g(x) = -x \\) \u3068\u306a\u308b\u3079\u304d\u3067\u306f\u3042\u308b\u304c\uff0c\u7a4d\u5206\u56e0\u5b50 \\(g(x)\\) \u306b\u306f\u5b9a\u6570\u500d\u306e\u4efb\u610f\u6027\u304c\u3042\u308b\u306e\u3067\uff0c\\( g(x) = x\\) \u3068\u3057\u3066\u3088\u3044\u3002<\/p>\n
\u3059\u308b\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\ny &=& \\frac{1}{g(x)} \\left\\{ \\int g(x) Q(x) dx + C\\right\\} \\\\
\n&=& \\frac{1}{x} \\left\\{ \\int x \\frac{\\sin x}{x} dx + C \\right\\}\\\\
\n&=& \\frac{1}{x} \\bigl\\{ -\\cos x + C\\bigr\\}
\n\\end{eqnarray}<\/p>\n
\u4ee5\u4e0b\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3002<\/p>\n
$$ \\frac{dy}{dx} -x\\, y = 2x $$<\/p>\n
\u7a4d\u5206\u56e0\u5b50\u6cd5\u3067\u89e3\u304f\u3002<\/p>\n
$$ P(x) = -x, \\qquad Q(x) = 2 x$$<\/p>\n
\u7a4d\u5206\u56e0\u5b50\u306f<\/p>\n
$$ g(x) = \\exp\\left\\{ \\int P(x) \\,dx \\right\\} =
\n\\exp\\left(-\\frac{x^2}{2}\\right) $$<\/p>\n
\u3053\u308c\u3092\u4f7f\u3046\u3068<\/p>\n
\\begin{eqnarray}
\ny(x) &=& \\frac{1}{g(x)} \\left\\{ \\int g(x) Q(x) dx + C \\right\\} \\\\\u00a0\u00a0 &=&
\n\\exp\\left(\\frac{x^2}{2}\\right) \\left\\{ \\int \\exp\\left(-\\frac{x^2}{2}\\right) 2x dx + C \\right\\} \\\\ &=&
\n-2 + C \\exp\\left(\\frac{x^2}{2}\\right)
\n\\end{eqnarray}<\/p>\n
\u3061\u306a\u307f\u306b\uff0c<\/p>\n
$$\\int \\exp\\left(-\\frac{x^2}{2}\\right) 2x dx$$<\/p>\n
\u306f\uff0c\\( t \\equiv x^2\\) \u3068\u3044\u3046\u5909\u6570\u5909\u63db\u3092\u3059\u308c\u3070\uff0c\\( dt = 2 x dx \\) \u3060\u304b\u3089\uff0c<\/p>\n
$$\\int^{x^2} \\exp\\left(-\\frac{t}{2}\\right) dt$$<\/p>\n
\u3068\u306a\u3063\u3066\u7c21\u5358\u306b\u7a4d\u5206\u3067\u304d\u308b\u3088\u306d\u3002<\/p>\n
\u4ee5\u4e0b\u306e\u3088\u3046\u306a\uff0c\u6307\u6570\u95a2\u6570\u3092\u542b\u3080\u7a4d\u5206\u306b\u3064\u3044\u3066\u5c11\u3057\u3002<\/p>\n
$$ \\int e^{ -a x^2}\\, 2 x dx $$<\/p>\n
\u5909\u6570\u5909\u63db<\/p>\n
$$\u00a0 x^2\u00a0 \\equiv t$$<\/p>\n
\u3068\u3057\u3066\u3084\u308c\u3070\uff0c<\/p>\n
$$2x dx = dt$$<\/p>\n
\u3068\u306a\u308b\u306e\u3067\uff0c<\/p>\n
$$ \\int e^{ -a x^2}\\, 2 x dx = \\int^{x^2} e^{ -a t} dt$$<\/p>\n
\u3053\u3053\u307e\u3067\u3084\u308b\u3068\uff0c\u7a4d\u5206\u3067\u304d\u305d\u3046\u3067\u3059\u3088\u306d\u3002<\/p>\n
\u3061\u306a\u307f\u306b\uff0c<\/p>\n
$$ \\int e^{ -a x^2}\u00a0 dx $$<\/p>\n
\u306f\u96e3\u3057\u3044\u3067\u3059\u3088\u3002\u3053\u306e\u4e0d\u5b9a\u7a4d\u5206\u306f\u89e3\u6790\u7684\u306b\u306f\u7a4d\u5206\u3067\u304d\u307e\u305b\u3093\u3002\u3057\u304b\u3057\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5b9a\u7a4d\u5206<\/p>\n
$$ \\int_0^{\\infty} e^{ -a x^2}\u00a0 dx $$<\/p>\n
\u306f\u89e3\u6790\u7684\u306b\u89e3\u3051\u308b\uff0c\u3068\u3044\u3046\u6975\u3081\u3066\u7279\u6b8a\u306a\u4f8b\uff08\u7e70\u308a\u8fd4\u3057\u307e\u3059\uff0c\u4e0d\u5b9a\u7a4d\u5206\u306f\u3067\u304d\u306a\u3044\u304c\u7121\u9650\u5927\u533a\u9593\u306e\u5b9a\u7a4d\u5206\u3060\u3051\u306f\u89e3\u6790\u7684\u306b\u89e3\u3051\u308b\uff01<\/strong><\/span>\uff09\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u3002\u3053\u308c\u306f\u30ac\u30a6\u30b9\u7a4d\u5206<\/strong><\/span>\u3068\u547c\u3070\u308c\u308b\u6709\u540d\u306a\u7a4d\u5206\u3067\uff0c\u3053\u306e\u6388\u696d\u3067\u3082\u3042\u3068\u3067\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\n \u4ee5\u4e0b\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3002\u3053\u3093\u306a\u554f\u984c\uff0c\u4eba\u751f\u306e\u3069\u3053\u3067\u4f7f\u3046\u3093\u3060\uff1f\u3068\u601d\u3046\u4eba\uff0c\u305d\u306e\u3046\u3061\u306b\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u306e\u5929\u4f53\u306e\u904b\u52d5\u3092\u8abf\u3079\u308b\u3068\u304d\u306b\u51fa\u3066\u304f\u308b\u3093\u3067\u3059\u3088<\/a>\u3002<\/p>\n $$2 \\sin x \\frac{dy}{dx} -2 y\\, \\cos x = \\cos x -\\cos^3 x $$<\/p>\n \u7a4d\u5206\u56e0\u5b50\u6cd5\u3067\u89e3\u304f\u306e\u3060\u3051\u3069\uff0c\u7c21\u5358\u3060\u304b\u3089\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3002\u4e21\u8fba\u3092 $2$ \u3067\u5272\u3063\u305f\u3042\u3068\u306e\u5de6\u8fba\u304c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u308b\u3053\u3068\u304c\u308f\u304b\u308c\u3070\u7c21\u5358\u3002<\/p>\n \\begin{eqnarray} <\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2253,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-6170","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t
\n\u4f8b\u984c 3<\/h3>\n
\n\\sin^2 x \\frac{d}{dx} \\left(\\frac{y}{\\sin x} \\right)&=& \\frac{\\cos x -\\cos^3 x}{2} \\\\
\n\\frac{d}{dx} \\left(\\frac{y}{\\sin x} \\right)&=&\\frac{\\cos x -\\cos^3 x}{2 \\sin^2 x} \\\\
\n\\frac{y}{\\sin x} &=& \\frac{1}{2} \\int \\frac{\\cos x -\\cos^3 x}{\\sin^2 x}\\, dx \\\\
\n&=& \\frac{1}{2} \\int \\frac{1 – \\cos^2 x}{\\sin^2 x}\\, \\cos x\\, dx \\\\
\n&=& \\frac{1}{2} \\sin x \\\\
\n\\therefore\\ \\ y &=& \\dots
\n\\end{eqnarray}<\/p>\n