\u3000\u3044\u304b\u306a\u308b 1 \u968e\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3082\u6b21\u306e\u5f62\u306b\u7f6e\u304d\u63db\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\uff0e<\/p>\n

$\\displaystyle \\frac{dy}{dx} = f(x,y)$ <\/p>\n

\u307e\u305f\u306f<\/p>\n

$\\displaystyle M(x,y)dx + N(x,y)dy = 0$ <\/p>\n

\u305d\u3057\u3066\u305d\u308c\u3089\u306e\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\u306f\u4e00\u3064\u306e\u4efb\u610f\u5b9a\u6570\u3092\u6301\u3061\u307e\u3059\uff0e\u69d8\u3005\u306a\u7a2e\u985e\u306e 1 \u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\u306e\u767a\u898b\u306b\u306f\u591a\u304f\u306e\u7279\u6b8a\u306a\u88c5\u7f6e\u304c\u6709\u7528\u3067\u3059\uff0e\u4e0b\u8a18\u306e\u30ea\u30b9\u30c8\u306b\u306f\u305d\u306e\u5e7e\u3064\u304b\u3092\u793a\u3057\u3066\u3042\u308a\u307e\u3059\uff0e<\/p>\n

\u5909\u6570\u5206\u96e2\u6cd5<\/li>\n
\u5b8c\u5168\u5fae\u5206\u65b9\u7a0b\u5f0f<\/li>\n
\u7a4d\u5206\u56e0\u5b50<\/li>\n
\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f<\/li>\n
\u540c\u6b21\u65b9\u7a0b\u5f0f<\/li>\n
\u30d9\u30eb\u30cc\u30fc\u30a4\u306e\u65b9\u7a0b\u5f0f<\/li>\n
y \u306b\u3064\u3044\u3066\u89e3\u3051\u308b\u65b9\u7a0b\u5f0f<\/li>\n
\u30af\u30ec\u30ed\u30fc\u306e\u65b9\u7a0b\u5f0f<\/li>\n
\u305d\u306e\u4ed6\u306e\u65b9\u7a0b\u5f0f<\/li>\n<\/ol>\n
1. \u5909\u6570\u5206\u96e2\u6cd5<\/h2>\n
\u3000\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u4e0b\u8a18\u306e\u3088\u3046\u3067\u3042\u308b\u5834\u5408\u306f<\/p>\n
$\\displaystyle f_1(x)g_1(y)dx + f_2(x)g_2(y)dy = 0$ <\/p>\n
\u4e00\u822c\u89e3\u3092\u5f97\u308b\u306b\u306f $g_1(y)f_2(x) \\ne 0$ \u3067\u9664\u3057\uff0c\u7a4d\u5206\u3057\u307e\u3059\uff0e<\/p>\n
$\\displaystyle \\int\\frac{f_1(x)}{f_2(x)}dx + \\int\\frac{g_2(y)}{g_1(y)}dy = c$ <\/p>\n
2. \u5b8c\u5168\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h2>\n
\u3000\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u4e0b\u8a18\u306e\u3088\u3046\u3067\u3042\u308b\u5834\u5408\u306f<\/p>\n
$\\displaystyle M(x, y)dx + N(x, y)dy = 0$ <\/p>\n
\u3053\u3053\u3067 $\\displaystyle \\frac{\\partial M}{\\partial y} = \\frac{\\partial N}{\\partial x}$ <\/p>\n
\u3000\u3053\u306e\u65b9\u7a0b\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u66f8\u304d\u63db\u3048\u3089\u308c\u307e\u3059\uff0e<\/p>\n
$\\displaystyle Mdx + Ndy = dU(x, y) = 0$ <\/p>\n
\u3053\u3053\u3067 dU \u306f\u5b8c\u5168\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u3059\uff0e\u3086\u3048\u306b\u305d\u306e\u89e3\u306f $U(x, y) = c$ \u307e\u305f\u306f\u540c\u7b49\u306e\u89e3\u3068\u3057\u3066<\/p>\n
$\\displaystyle \\int M\\partial x + \\int\\left(N - \\frac{\\partial}{\\partial y}\\int M\\partial x\\right)dy = c$ <\/p>\n
\u3053\u3053\u3067 δx \u306f y \u3092\u5b9a\u6570\u3068\u3057 x \u306b\u3088\u308b\u7a4d\u5206\u3092\u884c\u3046\u3053\u3068\u3092\u793a\u3057\u307e\u3059\uff0e<\/p>\n
3. \u7a4d\u5206\u56e0\u5b50<\/h2>\n
\u3000\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u4e0b\u8a18\u306e\u3088\u3046\u3067\u3042\u308b\u5834\u5408\uff0c<\/p>\n
$\\displaystyle M(x, y)dx + N(x, y)dy = 0$ <\/p>\n
\u3053\u3053\u3067<\/p>\n
$\\displaystyle \\frac{\\partial M}{\\partial y} \\neq \\frac{\\partial N}{\\partial x}$ <\/p>\n
\u3000\u3053\u306e\u65b9\u7a0b\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u5b8c\u5168\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u66f8\u304d\u63db\u3048\u3089\u308c\u307e\u3059\uff0e<\/p>\n
$\\displaystyle \\mu M dx + \\mu N dy = 0$ <\/p>\n
\u3053\u3053\u3067 μ \u306f\u9069\u5207\u306a\u7a4d\u5206\u56e0\u5b50\u3067\u3059\uff0e<\/p>\n
\u3000\u4e0b\u8a18\u306e\u7d44\u307f\u5408\u308f\u305b\u306f\u7a4d\u5206\u56e0\u5b50\u3092\u767a\u898b\u3059\u308b\u306e\u306b\u3057\u3070\u3057\u3070\u6709\u7528\u3067\u3059\uff0e<\/p>\n
$\\displaystyle \\frac{xdy - ydx}{x^2} = d\\left(\\frac{y}{x}\\right)$
\n $\\displaystyle \\frac{xdy - ydx}{y^2} = -d\\left(\\frac{x}{y}\\right)$
\n $\\displaystyle \\frac{xdy - ydx}{x^2 + y^2} = d\\left(\\tan^{-1}\\frac{y}{x}\\right)$
\n $\\displaystyle \\frac{xdy - ydx}{x^2 - y^2} = \\frac{1}{2}d\\left(\\ln\\frac{x - y}{x + y}\\right)$
\n $\\displaystyle \\frac{xdx + ydy}{x^2 + y^2} = \\frac{1}{2}d\\{\\ln(x^2 + y^2)\\}$ <\/p>\n
4. \u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h2>\n
\u3000\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u4e0b\u8a18\u306e\u3088\u3046\u3067\u3042\u308b\u5834\u5408\uff0c<\/p>\n
$\\displaystyle \\frac{dy}{dx} + P(x)y = Q(x)$ <\/p>\n
\u3000\u7a4d\u5206\u56e0\u5b50\u306f\u6b21\u306e\u3088\u3046\u306b\u5f97\u3089\u308c\u307e\u3059\uff0e<\/p>\n
$\\displaystyle \\mu = e^{\\int P(x)dx}$ <\/p>\n
\u307e\u305f\u65b9\u7a0b\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u66f8\u304d\u63db\u3048\u3089\u308c\u307e\u3059\uff0e<\/p>\n
$\\displaystyle \\frac{d}{dx}(\\mu y) = \\mu Q$ <\/p>\n
\u89e3\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u3067\u3059\uff0e<\/p>\n
$\\displaystyle \\mu y = \\int \\mu Qdx + c$ <\/p>\n
\u307e\u305f\u306f<\/p>\n
$\\displaystyle ye^{\\int Pdx} = \\int Qe^{\\int Pdx}dx + c$ <\/p>\n
5. \u540c\u6b21\u65b9\u7a0b\u5f0f<\/h2>\n
\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u4e0b\u8a18\u306e\u3088\u3046\u3067\u3042\u308b\u5834\u5408\uff0c<\/p>\n
$\\displaystyle \\frac{dy}{dx} = F\\left(\\frac{y}{x}\\right)$ <\/p>\n
\u3000 $y\/x = v$ \u307e\u305f\u306f $y = vx$ \u3068\u3057\u307e\u3059\uff0e\u3059\u308b\u3068\u65b9\u7a0b\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\uff0e<\/p>\n
$\\displaystyle v + x\\frac{dv}{dx} + F(x)$ <\/p>\n
\u307e\u305f\u306f<\/p>\n
$\\displaystyle xdv + (F(x) - v)dx = 0$ <\/p>\n
\u3053\u3053\u3067 Type 1 \u3068\u540c\u3058\u306b\u306a\u308a\uff0c\u89e3\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\uff0e<\/p>\n
$\\displaystyle \\ln x = \\int \\frac{dv}{F(v) - v} + c$ <\/p>\n
\u3053\u3053\u3067 $v = y\/x$ \u3067\u3059\uff0e\u3082\u3057 $F(v) = v$ \u306a\u3089\u89e3\u306f $y = cx$ \u3068\u306a\u308a\u307e\u3059\uff0e<\/p>\n
6. \u30d9\u30eb\u30cc\u30fc\u30a4\u306e\u65b9\u7a0b\u5f0f<\/h2>\n
\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u4e0b\u8a18\u306e\u3088\u3046\u3067\u3042\u308b\u5834\u5408\uff0c<\/p>\n
$\\displaystyle \\frac{dy}{dx} + P(x)y = Q(x)y^n,\\ n \\neq 0, 1$ <\/p>\n
\u3000 $v = y^{1 - n}$ \u3068\u3057\u307e\u3059\uff0e\u3059\u308b\u3068\u65b9\u7a0b\u5f0f\u306f Type 4 \u306b\u7f6e\u304d\u63db\u3048\u3089\u308c\uff0c\u89e3\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\uff0e<\/p>\n
$\\displaystyle ve^{(1-n)\\int Pdx} = (1 - n)\\int Qe^{(1-n)\\int Pdx}dx + c$ <\/p>\n
\u3000\u4eee\u306b n = 0 \u306a\u3089\u65b9\u7a0b\u5f0f\u306f Type 4 \u3068\u540c\u3058\u3067\u3042\u308a\uff0c n = 1 \u306a\u3089 Type 1 \u3068\u540c\u3058\u3067\u3059\uff0e<\/p>\n
7. y \u306b\u3064\u3044\u3066\u89e3\u3051\u308b\u65b9\u7a0b\u5f0f<\/h2>\n
\u3000\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u4e0b\u8a18\u306e\u3088\u3046\u3067\u3042\u308b\u5834\u5408\uff0c<\/p>\n
$\\displaystyle y = g(x, y)$ <\/p>\n
\u3053\u3053\u3067<\/p>\n
$\\displaystyle p = y'$ <\/p>\n
\u3000\u65b9\u7a0b\u5f0f\u306e\u4e21\u8fba\u3092 x \u306b\u3064\u3044\u3066\u5fae\u5206\u3059\u308b\u3068\u6b21\u304c\u5f97\u3089\u308c\u307e\u3059\uff0e<\/p>\n
$\\displaystyle \\frac{dy}{dx} = \\frac{dg}{dx} = \\frac{\\partial g}{\\partial x} + \\frac{\\partial g}{\\partial p}\\frac{\\partial p}{\\partial x}$ <\/p>\n
\u307e\u305f\u306f<\/p>\n
$\\displaystyle p = \\frac{\\partial g}{\\partial x} + \\frac{\\partial g}{\\partial p}\\frac{\\partial p}{\\partial x}$ <\/p>\n
\u3000\u305d\u3057\u3066\u3053\u306e\u6700\u5f8c\u306e\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3068 $G(x, p, c) = 0$ \u304c\u5f97\u3089\u308c\u307e\u3059\uff0e\u5fc5\u8981\u306a\u89e3\u306f $G(x, p, c) = 0$ \u3068 $y = g(x, p)$ \u306e\u9593\u306e p \u3092\u6d88\u53bb\u3057\u3066\u5f97\u3089\u308c\u307e\u3059\uff0e<\/p>\n
\u3000\u305d\u306e\u65b9\u7a0b\u5f0f\u3092 x \u306b\u3064\u3044\u3066\u89e3\u304f\u305f\u3081\u306e\u985e\u4f3c\u306e\u65b9\u6cd5\u304c\u5b58\u5728\u3057\u307e\u3059\uff0e<\/p>\n
8. \u30af\u30ec\u30ed\u30fc\u306e\u65b9\u7a0b\u5f0f<\/h2>\n
\u3000\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u4e0b\u8a18\u306e\u3088\u3046\u3067\u3042\u308b\u5834\u5408\uff0c<\/p>\n
$\\displaystyle y = px + F(p)$ <\/p>\n
\u3053\u3053\u3067<\/p>\n
$\\displaystyle p = y'$ <\/p>\n
\u3000\u3053\u306e\u65b9\u7a0b\u5f0f\u306f Type 7 \u306b\u7f6e\u304d\u63db\u3048\u3089\u308c\uff0c\u89e3\u306f\u6b21\u306e\u901a\u308a\u3067\u3059\uff0e<\/p>\n
$\\displaystyle y = cx + F(c)$ <\/p>\n
\u3000\u3053\u306e\u65b9\u7a0b\u5f0f\u3082\u307e\u305f\u4e00\u822c\u89e3\u306e\u3046\u3061\u5358\u89e3\u3092\u6709\u3057\u307e\u3059\uff0e<\/p>\n
9. \u305d\u306e\u4ed6\u306e\u65b9\u7a0b\u5f0f<\/h2>\n
\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u4e0b\u8a18\u306e\u3088\u3046\u3067\u3042\u308b\u5834\u5408\uff0c<\/p>\n
$\\displaystyle (a) \\frac{dy}{dx} = F(\\alpha x + \\beta y)\\\\\\vspace{0.2 in} (b) \\frac{dy}{dx} = F\\left(\\frac{\\alpha_1 x + \\beta_1 y + \\gamma_1}{\\alpha_2 x + \\beta_2 y + \\gamma_2}\\right)$ <\/p>\n
(a) $\\alpha x + \\beta y = v$ \u3068\u3059\u308b\u3068\uff0c\u65b9\u7a0b\u5f0f\u306f Type 1 \u306b\u7f6e\u304d\u63db\u3048\u3089\u308c\u307e\u3059\uff0e<\/p>\n
(b) $x = X +h,\\ y = Y + k$ \u3068\u3057\uff0c\u5b9a\u6570 h \u3068 k \u3092\u65b9\u7a0b\u5f0f\u304c Type 5 \u306b\u7f6e\u304d\u63db\u3048\u3089\u308c\u308b\u3088\u3046\u306b\u5b9a\u3081\u307e\u3059\uff0e\u3053\u306e\u51e6\u7406\u306f\u4ee5\u4e0b\u306e\u5834\u5408\uff0c\u3059\u306a\u308f\u3061 $\\alpha_1\/\\alpha_2 \\neq \\beta_1\/\\beta_2$ \u306e\u5834\u5408\u306b\u9650\u308a\u53ef\u80fd\u3067\u3059\uff0e\u4eee\u306b $\\alpha_1\/\\alpha_2 = \\beta_1\/\\beta_2$ \u306e\u5834\u5408\uff0c\u65b9\u7a0b\u5f0f\u306f Type 9(a) \u306b\u7f6e\u304d\u63db\u3048\u3089\u308c\u307e\u3059\uff0e<\/p>\n
<\/iframe><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>\u3000\u3044\u304b\u306a\u308b 1 \u968e\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3082\u6b21\u306e\u5f62\u306b\u7f6e\u304d\u63db\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\uff0e \u307e\u305f\u306f \u305d\u3057\u3066\u305d\u308c\u3089\u306e\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\u306f\u4e00\u3064\u306e\u4efb\u610f\u5b9a\u6570\u3092\u6301\u3061\u307e\u3059\uff0e\u69d8\u3005\u306a\u7a2e\u985e\u306e 1 \u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\u306e\u767a\u898b\u306b\u306f\u591a\u304f\u306e\u7279\u6b8a\u306a\u88c5\u7f6e\u304c\u6709\u7528\u3067\u3059\uff0e\u4e0b\u8a18\u306e\u30ea\u30b9\u30c8 … <a href=\"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=4589\" class=\"more-link\"><span class=\"screen-reader-text\">“\u7279\u6b8a\u306a1\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3068\u305d\u306e\u89e3” \u306e<\/span>\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":1,"featured_media":6040,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[9],"tags":[],"class_list":["post-4589","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/4589","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4589"}],"version-history":[{"count":15,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/4589\/revisions"}],"predecessor-version":[{"id":6795,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/posts\/4589\/revisions\/6795"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=\/wp\/v2\/media\/6040"}],"wp:attachment":[{"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4589"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4589"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fujiitoshiki.com\/improvesociety\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4589"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}