\u300c\u975e\u7dda\u5f62\u300d\u3068\u3044\u3063\u3066\u3082\uff0c\u672a\u77e5\u95a2\u6570 $y(x)$ \u304a\u3088\u3073\u305d\u306e1\u968e\u5fae\u5206 $\\frac{dy}{dx}$ \u306b\u3064\u3044\u3066\u4e8c\u4e57\u306e\u9805\u306e\u307f\u3092\u542b\u3080\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a1\u968e\u975e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3002<\/p>\n
$$\\left( \\frac{dy}{dx}\\right)^2\u00a0 = 1 -y^2$$<\/p>\n
\u3053\u306e\u5f62\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f\uff0c\u30cb\u30e5\u30fc\u30c8\u30f3\u529b\u5b66\u306b\u304a\u3051\u308b\u30b1\u30d7\u30e9\u30fc\u554f\u984c\u3084\uff0c\u4e00\u822c\u76f8\u5bfe\u8ad6\u306b\u304a\u3044\u3066\u30b7\u30e5\u30d0\u30eb\u30c4\u30b7\u30eb\u30c8\u6642\u7a7a\u4e2d\u306e\u5149\u306e\u4f1d\u64ad\u3084\u5929\u4f53\u306e\u904b\u52d5\u3092\u89e3\u304f\u3068\u304d\u306b\u3088\u304f\u51fa\u3066\u304f\u308b\u306e\u3067\uff0c\u3057\u3063\u304b\u308a\u3084\u3063\u3066\u304a\u3053\u3046\u3002<\/p>\n
\\begin{eqnarray}
\n\\left( \\frac{dy}{dx}\\right)^2\u00a0 &=& 1 -y^2 \\\\
\n\\frac{dy}{dx} &=& \\pm \\sqrt{1 -y^2} \\\\
\n\\frac{dy}{\\sqrt{1 -y^2}} &=& \\pm dx \\\\
\n\\int \\frac{dy}{\\sqrt{1 -y^2}} &=& \\pm \\int dx \\\\
\n\\sin^{-1} y &=& \\pm x + C \\\\
\n\\therefore\\ \\ y &=& \\sin\\left(\\pm x + C \\right)
\n\\end{eqnarray}<\/p>\n
\u9006\u4e09\u89d2\u95a2\u6570\u306e\u5fae\u5206<\/strong><\/span><\/a> $\\displaystyle \\left( \\sin^{-1} x\\right)’ = \\frac{1}{\\sqrt{1-x^2}}$ \u306a\u3093\u3066\uff0c\u3044\u3063\u305f\u3044\u3069\u3053\u3067\u4f7f\u3046\u3093\u3060\u3068\u304b\u601d\u3063\u3066\u3044\u305f\u3067\u3057\u3087\uff1f\u3000\u3053\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u305f\u3081\u306b\u4f7f\u3046\u3093\u3067\u3059\u3088\uff01<\/p>\n \u8907\u53f7 $+$ \u306e\u5834\u5408\u306f\u305d\u306e\u307e\u307e<\/p>\n $$y = \\sin(x + C)$$<\/p>\n \u8907\u53f7 $-$ \u306e\u5834\u5408\u306f\uff0c$C \\rightarrow \\pi -C’$ \u3068\u304a\u3044\u3066<\/p>\n \\begin{eqnarray} \u3057\u305f\u304c\u3063\u3066\u7a4d\u5206\u5b9a\u6570\u3092\u3042\u3089\u305f\u3081\u3066 $C$ \u3068\u304a\u304d\u306a\u304a\u3057\u3066\u307f\u308b\u3068\u7b54\u3048\u306f<\/p>\n $$y = \\sin(x + C)$$<\/p>\n \u3068\u3057\u3066\u3088\u3044\u3053\u3068\u306b\u306a\u308b\u3002\u975e\u7dda\u5f62\u3067\u306f\u3042\u308b\u304c\uff0c1\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306a\u306e\u3067\u7a4d\u5206\u5b9a\u6570\u306f1\u500b\u3002\u3053\u308c\u306f\u521d\u671f\u6761\u4ef6\u30921\u3064\u8ab2\u3057\u3066\u6c7a\u5b9a\u3059\u308b\u3053\u3068\u306b\u306a\u308b\u3002\u3042\u308b\u3044\u306f\uff0c$C = \\frac{\\pi}{2} + C^{\\prime\\prime}$ \u3068\u3057\u3066<\/p>\n \\begin{eqnarray} \u3053\u306e\u307e\u307e\u3067\u3082\u4e0a\u8a18\u306e\u3088\u3046\u306b\u89e3\u3051\u305f\u308f\u3051\u3060\u304c\uff0c\u3053\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4e21\u8fba\u3092 $x$ \u3067\u5fae\u5206\u3059\u308b\u3068\uff0c\u898b\u6163\u308c\u305f\u5f62\u306b\u306a\u308b\u3002<\/p>\n \\begin{eqnarray}
\ny &=& \\sin(-x + C) \\\\
\n&=& \\sin \\left( -x + \\pi\u00a0 -C’ \\right)\\\\
\n&=& \\sin \\left( \\pi -\\left(x + C’\\right) \\right)\\\\
\n&=& \\sin \\left(x + C’\\right)
\n\\end{eqnarray}<\/p>\n
\ny &=& \\sin(x + C) \\\\
\n&=& \\sin( \\frac{\\pi}{2} +x +\u00a0 C^{\\prime\\prime}) \\\\
\n&=& \\cos(x +\u00a0 C^{\\prime\\prime})
\n\\end{eqnarray}<\/p>\n2\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u3057\u3066\u89e3\u304f\u4f8b<\/h3>\n
\n\\frac{d}{dx} \\left( \\frac{dy}{dx}\\right)^2\u00a0 &=& \\frac{d}{dx}\\left(1 -y^2 \\right) \\\\
\n2 \\frac{dy}{dx} \\frac{d^2y}{dx^2} &=& -2 y \\frac{dy}{dx} \\\\
\n\\therefore\\ \\ \\frac{d^2y}{dx^2} &=& -y
\n\\end{eqnarray}<\/p>\n