{"id":2253,"date":"2022-02-24T18:14:24","date_gmt":"2022-02-24T09:14:24","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2253"},"modified":"2025-05-05T11:39:56","modified_gmt":"2025-05-05T02:39:56","slug":"1%e9%9a%8e%e7%b7%9a%e5%bd%a2%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%a8%e7%a9%8d%e5%88%86%e5%9b%a0%e5%ad%90%e6%b3%95","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/%e5%b8%b8%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%bc%8f\/1%e9%9a%8e%e7%b7%9a%e5%bd%a2%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%a8%e7%a9%8d%e5%88%86%e5%9b%a0%e5%ad%90%e6%b3%95\/","title":{"rendered":"1\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\u3068\u7a4d\u5206\u56e0\u5b50\u6cd5"},"content":{"rendered":"

\u4e00\u822c\u7684\u306a1\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5f62\u3002<\/p>\n

$$
\n\\frac{dy}{dx} + P(x)\\, y = Q(x)
\n$$
\n<\/p>\n

\u672a\u77e5\u95a2\u6570\uff08\u3053\u308c\u304b\u3089\u6c42\u3081\u308b\u95a2\u6570\uff09\\(y(x)\\) \u3092\u542b\u3080\u9805\u3092\u5168\u3066\u5de6\u8fba\u306b\u3082\u3063\u3066\u3044\u3063\u3066\u3082\uff0c\u53f3\u8fba\u304c\u30bc\u30ed\u3068\u306a\u3063\u3066\u3044\u306a\u3044\u300c\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u300d\u3002<\/p>\n

\u7279\u306b\uff0c\u53f3\u8fba\uff0c\\(Q(x) = 0\\) \u306a\u3089\u300c\u540c\u6b21\u65b9\u7a0b\u5f0f\u300d\uff1a<\/p>\n

$$
\n\\frac{dy}{dx} + P(x)\\, y = 0
\n$$<\/p>\n

\u3053\u306e\u5f62\u306a\u3089\uff0c\u5909\u6570\u5206\u96e2\u6cd5\u3067\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u305f\u3088\u306d\uff01<\/p>\n

\u7a4d\u5206\u56e0\u5b50\u6cd5<\/h3>\n

\u4e00\u822c\u7684\u306a1\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\uff08\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u683c\u597d\u3057\u3066\u3044\u308b\u3082\u306e\uff09<\/p>\n

$$
\n\\frac{dy}{dx} + P(x)\\, y = Q(x)
\n$$<\/p>\n

\u306e\u4e21\u8fba\u306b\u9069\u5f53\u306a\u95a2\u6570 \\(g(x)\\) \u3092\u304b\u3051\u308b\u3068<\/p>\n

$$
\ng(x) \\left( \\frac{dy}{dx} + P(x)\\, y \\right) = g(x)\\,Q(x)
\n$$<\/p>\n

\u3053\u306e \\(g(x) \\) \u3092\u3046\u307e\u304f\u3068\u3063\u3066\uff0c\u5de6\u8fba\u304c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5f62\uff08\u306a\u306b\u304b\u306e\u95a2\u6570\u306e\u639b\u3051\u7b97\u3067\u3082\u3044\u3044\u304b\u3089\u3001\u3072\u3068\u304b\u305f\u307e\u308a\u3068\u306a\u3063\u305f\u5168\u4f53\u3092 \\(x\\) \u3067\u5fae\u5206\u3057\u305f\u5f62\uff09\u306b\u306a\u308b\u3088\u3046\u306b\u3059\u308b\u3002<\/p>\n

$$
\ng(x) \\left( \\frac{dy}{dx} + P(x)\\, y \\right) = \\frac{d}{dx} \\bigl(g(x)\\, y\\bigr)
\n$$<\/p>\n

\u305d\u3046\u3059\u308b\u3068\uff0c\u3082\u3068\u3082\u3068\u306e\u5f0f\u306f<\/p>\n

$$
\n\\frac{d}{dx} \\bigl(g(x)\\, y\\bigr)= g(x)\\,Q(x)
\n$$<\/p>\n

\u3068\u306a\u308a\uff0c\u3053\u308c\u306f\u4e21\u8fba\u3092\u7a4d\u5206\u3059\u308c\u3070 \\(g(x)\\, y \\) \u306b\u3064\u3044\u3066\u89e3\u3051\uff0c\u3057\u305f\u304c\u3063\u3066 \\(y \\) \u306b\u3064\u3044\u3066\u3082\u7c21\u5358\u306b\u6c42\u3081\u3089\u308c\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

\u3055\u3066\u3001\\( g(x) \\) \u306f\u4ee5\u4e0b\u306e\u5f0f\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u6c7a\u3081\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

$$
\ng(x) \\left( \\frac{dy}{dx} + P(x)\\, y \\right) = \\frac{d}{dx} \\bigl(g(x)\\, y\\bigr)
\n$$<\/p>\n

\u3053\u306e\u4e21\u8fba\u3092\u3042\u3089\u305f\u3081\u3066\u8a08\u7b97\u3057\u3066\u307f\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray}
\ng(x) \\left( \\frac{dy}{dx} + P(x)\\, y \\right) &=&
\ng(x) \\frac{dy}{dx} + g(x)\\,P(x)\\, y \\\\
\n\\frac{d}{dx} \\bigl(g(x)\\, y\\bigr) &=& g(x) \\frac{dy}{dx} + \\frac{dg}{dx} y
\n\\end{eqnarray}<\/p>\n

\u3067\u3042\u308b\u304b\u3089\uff0c\u4e21\u8fba\u3092\u307f\u304f\u3089\u3079\u308b\u3068\uff0c\\(g(x)\\) \u306f<\/p>\n

$$ \\frac{dg}{dx} = P(x)\\, g(x)
\n$$<\/p>\n

\u3092\u6e80\u305f\u3055\u306a\u3044\u3068\u3044\u3051\u306a\u3044\u3002\u3053\u308c\u306f \\(g(x)\\) \u306b\u3064\u3044\u3066\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u3042\u308a\uff0c\u5909\u6570\u5206\u96e2\u6cd5\u3092\u4f7f\u3063\u3066\u6b21\u306e\u3088\u3046\u306b\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n

$$\\frac{dg}{g} = P(x)\\, dx$$<\/p>\n

$$\\int \\frac{dg}{g} = \\int P(x)\\, dx$$<\/p>\n

$$ \\ln |g| = \\int P(x)\\, dx$$<\/p>\n

$$ |g| = \\exp\\left\\{ \\int P(x)\\, dx\\right\\}$$<\/p>\n

\\(g > 0\\) \u306e\u5834\u5408\u3092\u8003\u3048\u308c\u3070\u5341\u5206\u3067\u3042\u308b\u306e\u3067\uff08\u306a\u305c\u304b\u3063\u3066\uff1f\u4ee5\u4e0b\u306e\u88dc\u8db3\u3092\u53c2\u7167\u3002\uff09<\/p>\n

$$ g(x) = \\exp\\left\\{ \\int P(x)\\, dx \\right\\} $$
\n\\(\\displaystyle \\int P(x)\\, dx\\)\u00a0 \u306f\u4e0d\u5b9a\u7a4d\u5206\u3067\u3042\u308b\u304b\u3089\uff0c\u4efb\u610f\u306e\u7a4d\u5206\u5b9a\u6570\u3092\u3064\u3051\u308b\u3079\u304d\uff0c\u3068\u601d\u3046\u304b\u3082\u3057\u308c\u306a\u3044\u304c\uff0c\u4ee5\u4e0b\u306e\u88dc\u8db3\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b \\(g(x)\\) \u3092\u6c42\u3081\u308b\u969b\u306b\u306f\u7a4d\u5206\u5b9a\u6570\u306f\u7701\u7565\u3057\u3066\u5927\u4e08\u592b\u3067\u3059\u3088\u3002<\/p>\n

\u7a4d\u5206\u56e0\u5b50<\/b><\/span>\u3068\u547c\u3070\u308c\u308b\u3053\u306e\u95a2\u6570 \\(g(x) \\) \u304c\u308f\u304b\u308b\u3068\uff0c\u3082\u3068\u306e1\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f<\/p>\n

$$\\frac{d}{dx} \\bigl(g(x)\\, y\\bigr) = g(x)\\,Q(x) $$<\/p>\n

\u3067\u3042\u308b\u304b\u3089\uff0c\u4e21\u8fba\u3092 \\(x\\) \u3067\u7a4d\u5206\u3057\u305f\u306e\u3061\u306b \\(g(x) \\) \u3067\u308f\u3063\u3066\uff0c<\/p>\n

$$ y = \\frac{1}{g(x)} \\left\\{ \\int g(x)\\,Q(x)\\, dx + C \\right\\} $$<\/p>\n

\u3068\u306a\u308b\u3002\u4ee5\u4e0a\u304c\u7a4d\u5206\u56e0\u5b50\u6cd5\u306b\u3088\u308b1\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\u3067\u3042\u308b\u3002\u4e0d\u5b9a\u7a4d\u5206 \\(\\displaystyle\u00a0 \\int g(x)\\,Q(x)\\, dx\\) \u306f\u3053\u308c\u81ea\u4f53\u7a4d\u5206\u5b9a\u6570\u3092\u542b\u3080\u3067\u3042\u308b\u304c\uff0c\u3053\u3053\u3067\u306f\u5225\u9014\u3042\u304b\u3089\u3055\u307e\u306b\u7a4d\u5206\u5b9a\u6570 $C$ \u3092\u8ffd\u8a18\u3057\u3066\u304a\u304f\u3002<\/p>\n

\u7a4d\u5206\u56e0\u5b50\u6cd5\u306e\u307e\u3068\u3081<\/h3>\n

1\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\uff08\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u683c\u597d\u3057\u3066\u3044\u308b\u3082\u306e\uff09<\/p>\n

$$
\n\\frac{dy}{dx} + P(x)\\, y = Q(x)
\n$$<\/p>\n

\u306e\u89e3\u306f<\/p>\n

$$ g(x) = \\exp\\left\\{ \\int P(x)\\, dx \\right\\} $$<\/p>\n

\u3067\u4e0e\u3048\u308c\u3089\u308b\u7a4d\u5206\u56e0\u5b50 \\(g(x)\\) \u3092\u4f7f\u3063\u3066<\/p>\n

$$ y = \\frac{1}{g(x)} \\left\\{ \\int g(x)\\,Q(x)\\, dx + C \\right\\} $$<\/p>\n

\u3068\u66f8\u3051\u308b\u3002$C$ \u306f\u7a4d\u5206\u5b9a\u6570\uff081\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3060\u304b\u3089\u7a4d\u5206\u5b9a\u6570\u306f1\u500b\uff09\u3002<\/p>\n

\u306a\u304a\uff0c\\(\\displaystyle \\int P(x)\\, dx\\) \u3084 \\( \\int g(x)\\,Q(x)\\, dx\\) \u306f\u4e0d\u5b9a\u7a4d\u5206\u306a\u306e\u3067\u3053\u308c\u81ea\u4f53\u306b\u7a4d\u5206\u5b9a\u6570\u3092\u542b\u3080\u306e\u3067\u3042\u308b\u304c\uff0c\u4ee5\u4e0b\u306e\u88dc\u8db3\u306b\u8ff0\u3079\u308b\u3088\u3046\u306a\u4efb\u610f\u6027\u306b\u3088\u3063\u3066\u7a4d\u5206\u5b9a\u6570\u306f\u7701\u7565\u3057\u3066\u3088\u3044<\/strong><\/span>\u3002\u3053\u306e\u3053\u3068\u3092\u3042\u304b\u3089\u3055\u307e\u306b\u8868\u3059\u305f\u3081\u306b\uff0c<\/p>\n

$$ \\int P(x)\\, dx = \\int^x P(x’)\\, dx’ + C$$<\/p>\n

\u306e\u3088\u3046\u306b\u7a4d\u5206\u5b9a\u6570\u3092\u7701\u7565\u3057\u305f\u90e8\u5206\u3092 \\(\\displaystyle \\int^x P(x’)\\, dx’\\) \u306e\u3088\u3046\u306b\u66f8\u304d\u5206\u3051\u3066\u307f\u305f\u304c\uff0c\uff08\u306a\u305c $x$ \u3058\u3083\u306a\u304f\u3066 $x’$ \u306a\u3093\u3067\u3059\u304b\u306a\u3069\u3068\u3044\u3046\u8cea\u554f\u304c\u6765\u305f\u308a\u3057\u3066\uff09\u5b66\u751f\u306b\u306f\u4e0d\u8a55\u3060\u3063\u305f\u306e\u3067\uff0c\u6df7\u4e71\u3055\u305b\u306a\u3044\u3088\u3046\u306b\u66f8\u304d\u65b9\u3092\u3082\u3068\u306b\u623b\u3057\u3066\u307f\u305f\u3002<\/p>\n

\u88dc\u8db3\uff1a\u7a4d\u5206\u56e0\u5b50\u306b\u304a\u3051\u308b\u5b9a\u6570\u500d\u306e\u4efb\u610f\u6027\u306b\u3064\u3044\u3066<\/h3>\n

\u7a4d\u5206\u56e0\u5b50 \\(g(x)\\) \u306e\u5f79\u5272\u3068\u306f\uff0c\u5fae\u5206\u65b9\u7a0b\u5f0f<\/p>\n

$$\\frac{dy}{dx} + P(x) y = Q(x)$$<\/p>\n

\u306e\u4e21\u8fba\u306b\\(g(x)\\)\u3092\u304b\u3051\u3066<\/p>\n

$$\\frac{d}{dx}\\bigl( g(x) \\,y(x) \\bigr) = g(x)\\, Q(x) $$<\/p>\n

\u306e\u5f62\u306b\u3059\u308b\uff0c\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u3063\u305f\u3002\u4eca\uff0c\u3053\u306e\u5f0f\u306e\u4e21\u8fba\u306b\u5b9a\u6570 \\( K\\) \u3092\u304b\u3051\u308b\u3068<\/p>\n

$$\\frac{d}{dx}\\bigl( K g(x) \\,y(x) \\bigr) = K g(x)\\, Q(x) $$<\/p>\n

\u3053\u308c\u306f\uff0c\\( \\tilde{g}(x) \\equiv K g(x) \\) \u3067\u5b9a\u7fa9\u3055\u308c\u308b \\(\\tilde{g}(x)\\) \u3082\u307e\u305f\u7a4d\u5206\u56e0\u5b50\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u3092\u610f\u5473\u3059\u308b\uff1a<\/p>\n

$$\\frac{d}{dx}\\bigl( \\tilde{g}(x)\\, y(x) \\bigr) = \\tilde{g}(x)\\, Q(x) $$<\/p>\n

\u3064\u307e\u308a\uff0c\u7a4d\u5206\u56e0\u5b50\u306f\u552f\u4e00\u7121\u4e8c\u306b\u6c7a\u307e\u308b\u306e\u3067\u306f\u306a\u304f\uff0c\u4efb\u610f\u306e\u5b9a\u6570\u3092\u304b\u3051\u3066\u3082\u3088\u3044\u3053\u3068\u306b\u306a\u308b\u3002\u3053\u308c\u3092\u696d\u754c\u7528\u8a9e\u3067\uff0c\u300c\u7a4d\u5206\u56e0\u5b50\u306b\u306f\u5b9a\u6570\u500d\u306e\u4efb\u610f\u6027\u304c\u3042\u308b\u300d\u3068\u3044\u3046\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u4e00\u822c\u7684\u306a1\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5f62\u3002<\/p>\n

$$ \\frac{dy}{dx} + P(x)\\, y = Q(x) $$<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":2224,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2253","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\/2253","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\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=2253"}],"version-history":[{"count":7,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2253\/revisions"}],"predecessor-version":[{"id":10217,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2253\/revisions\/10217"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2224"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2253"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}