<\/p>\n
\\( y^{”} + y = 0\\) \u3084 \\(y^{”} – y = 0\\) \u3092\u810a\u9ac4\u53cd\u5c04\u306b\u3088\u3089\u305a\u306b\u30b7\u30b9\u30c6\u30de\u30c6\u30a3\u30c3\u30af\u306b\u89e3\u304f\u3002\uff08\u9762\u5012\u3060\u3051\u3069\uff0c\u3053\u306e\u3088\u3046\u306b\u89e3\u3051\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3002\uff09<\/p>\n
\\( y^{”} + y\u00a0 = 0\\) \u306e\u4e21\u8fba\u306b \\(2y’\\) \u3092\u304b\u3051\u308b\u3068<\/p>\n
$$ 2 y’ y^{”} + 2 y y’ = 0$$<\/p>\n
\u3053\u308c\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<\/p>\n
$$\\left( \\left( y’\\right)^2 + y^2 \\right)’ = 0$$<\/p>\n
\u5fae\u5206\u3057\u3066\u30bc\u30ed\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u304b\u3063\u3053\u306e\u4e2d\u8eab\u306f\u5b9a\u6570\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u306a\u306e\u3067\uff0c<\/p>\n
$$\\left( y’\\right)^2 + y^2 = \\mbox{const.} \\equiv\u00a0 a^2$$<\/p>\n
\u3068\u304a\u3051\u308b\u3002\u3064\u307e\u308a\uff0c<\/p>\n
$$ \\frac{dy}{dx} = \\pm \\sqrt{a^2 – y^2}$$<\/p>\n
\u3092\u89e3\u3051\u3070\u3088\u3044\u3002\u3053\u308c\u306f\uff0c\u5909\u6570\u5206\u96e2\u6cd5\u3067\u89e3\u3051\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\frac{dy}{\\sqrt{a^2\u00a0 -y^2}} &=& \\pm dx\\\\
\n\\int\\frac{dy}{\\sqrt{a^2\u00a0 -y^2}} &=& \\pm \\int dx\\\\
\n\\sin^{-1} \\frac{y}{a} &=& \\pm x+ C\\\\
\n\\therefore\\ \\ y &=& a \\sin (\\pm x+ C) \\\\
\n&=& a \\left( \\pm \\sin x \\cos C + \\cos x \\sin C \\right) \\\\
\n&=& A \\cos x + B \\sin x
\n\\end{eqnarray}<\/p>\n
\u3053\u3053\u3067\uff0c$A \\equiv a \\sin C, \\ B \\equiv \\pm a \\cos C$ \u3068\u304a\u3044\u305f\u3002<\/p>\n
<\/p>\n
\u9762\u5012\u306a\u3053\u3068\u3092\u907f\u3051\u308b\u305f\u3081\u306b\u7c21\u7565\u5316\u30d0\u30fc\u30b8\u30e7\u30f3\u3067\u3002\\( y^{”} -y = 0\\) \u306e\u4e21\u8fba\u3092 \\( i^2 = -1\\) \u3067\u5272\u308b\u3068\uff0c<\/p>\n
$$\\frac{d^2 y}{d(i x)^2} + y = 0$$<\/p>\n
\u3064\u307e\u308a\uff0c\\( y^{”} -y = 0\\) \u306e\u89e3\u306f\uff0c\\( y^{”} + y = 0\\) \u306e\u89e3\u3067 $x \\rightarrow i x$ \u3068\u3044\u3046\u7f6e\u304d\u63db\u3048\u3092\u3059\u308c\u3070\u3044\u3044\u304b\u3089\uff0c<\/p>\n
\\begin{eqnarray}
\ny &=& A \\cos (i x) + B \\sin (i x) \\\\
\n&=& A \\cosh x + i B \\sinh x\\\\
\n&=& A \\cosh x + B’ \\sinh x
\n\\end{eqnarray}<\/p>\n
\u7c21\u7565\u5316\u3057\u306a\u3044\u30d0\u30fc\u30b8\u30e7\u30f3\u3067\u3084\u308b\u306a\u3089\u3070\uff0c\\( y^{”} -y\u00a0 = 0\\) \u306e\u4e21\u8fba\u306b \\(2y’\\) \u3092\u304b\u3051\u308b\u3068<\/p>\n
$$ 2 y’ y^{”} – 2 y y’ = 0$$<\/p>\n
\u3053\u308c\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\u3002<\/p>\n
$$\\left( \\left( y’\\right)^2 – y^2 \\right)’ = 0$$<\/p>\n
\u5fae\u5206\u3057\u3066\u30bc\u30ed\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u304b\u3063\u3053\u306e\u4e2d\u8eab\u306f\u5b9a\u6570\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u306a\u306e\u3067\uff0c<\/p>\n
$$\\left( y’\\right)^2 – y^2 = \\mbox{const.} \\equiv\u00a0 C$$<\/p>\n
\u3068\u304a\u3051\u308b\u3002\u3064\u307e\u308a\uff0c<\/p>\n
$$ \\frac{dy}{dx} = \\pm \\sqrt{y^2+C}$$<\/p>\n
\u3092\u89e3\u3051\u3070\u3088\u3044\u3002\u3053\u308c\u306f\uff0c\u5909\u6570\u5206\u96e2\u6cd5\u3067\u89e3\u3051\u308b\u3002\\(C > 0\\) \u306a\u3089\u3070 \\(C \\equiv a^2\\) \u3068\u304a\u3044\u3066<\/p>\n
\\begin{eqnarray}
\n\\frac{dy}{\\sqrt{y^2 + a^2}} &=& \\pm dx\\\\
\n\\int\\frac{dy}{\\sqrt{y^2 + a^2}} &=& \\pm \\int dx\\\\
\n\\sinh^{-1} \\frac{y}{a} &=& \\pm x+ C\\\\
\n\\therefore\\ \\ y &=& a \\sinh (\\pm x+ C) \\\\
\n&=& a \\left( \\pm \\sinh x \\cosh C + \\cosh x \\sinh C \\right) \\\\
\n&=& A \\cosh x + B \\sinh x
\n\\end{eqnarray}<\/p>\n
\u3053\u3053\u3067\uff0c$A \\equiv a \\sinh C, \\ B \\equiv \\pm a \\cosh C$ \u3068\u304a\u3044\u305f\u3002<\/p>\n
\\(C < 0\\) \u306a\u3089\u3070 \\( C \\equiv\u00a0 – a^2\\) \u3068\u304a\u3044\u3066<\/p>\n
\\begin{eqnarray}
\n\\frac{dy}{\\sqrt{y^2 – a^2}} &=& \\pm dx\\\\
\n\\int\\frac{dy}{\\sqrt{y^2 – a^2}} &=& \\pm \\int dx\\\\
\n\\cosh^{-1} \\frac{y}{a} &=& \\pm x+ C\\\\
\n\\therefore\\ \\ y &=& a \\cosh (\\pm x+ C) \\\\
\n&=& a \\left( \\cosh x \\cosh C \\pm \\sinh x \\sinh C \\right) \\\\
\n&=& A \\cosh x + B \\sinh x
\n\\end{eqnarray}<\/p>\n
\u3053\u3053\u3067\uff0c$A \\equiv a \\cosh C, \\ B \\equiv \\pm a \\sinh C$ \u3068\u304a\u3044\u305f\u3002<\/p>\n
\u3068\u3044\u3046\u308f\u3051\u3067\uff0c\\( y^{”} + y = 0\\) \u3084 \\(y^{”} -y = 0\\) \u3092\u810a\u9ac4\u53cd\u5c04\u306b\u3088\u3089\u305a\u306b\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u305f\u304c\uff0c\u4ee5\u5f8c\u306f\u9762\u5012\u306a\u306e\u3067\uff0c\u810a\u9ac4\u53cd\u5c04\u3067\u7b54\u3048\u3092\u53eb\u3076\u3088\u3046\u306b\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n
\u3053\u3053\u3067\u4f7f\u3063\u305f\u5fae\u5206\u7a4d\u5206\u306e\u516c\u5f0f\u306f\uff0c\u7406\u5de5\u7cfb\u306e\u6570\u5b66B\u3067\u3084\u3063\u3066\u3044\u308b\u3002\u4f8b\u3048\u3070\u4ee5\u4e0b\u3092\u53c2\u7167\uff1a<\/p>\n