\u6700\u3082\u7c21\u5358\u306a\u5b9a\u6570\u4fc2\u65702\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\uff1a\u7d9a\u304d<\/a>\u300d\uff09<\/p>\n\\(f(x)\\) \u304a\u3088\u3073 \\(K\\) \u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u6c42\u3081\u308b\u3002<\/p>\n
\u307e\u305a\uff0c$\\bigl\\{f(x) y \\bigr\\}^{”} + K \\bigl\\{f(x) y \\bigr\\}$ \u3092\u5c55\u958b\u3057\u3066<\/p>\n
$$\\bigl\\{f(x) y \\bigr\\}^{”} + K \\bigl\\{f(x) y \\bigr\\} = f(x) y^{”} + 2 f'(x) y’ + f^{”}(x) y + K f(x) y
\n$$<\/p>\n
\u307e\u305f\uff0c\\( y^{”} + 2 b\\, y’ + c\\, y = 0 \\) \u306e\u5de6\u8fba\u306b\u95a2\u6570 \\(f(x)\\) \u3092\u304b\u3051\u305f\u3082\u306e\u306f<\/p>\n
$$ f(x) y^{”} + 2 b f(x) y’ + c f(x) y
\n$$<\/p>\n
\u3053\u308c\u3089\u304c\u7b49\u3057\u304f\u306a\u308b\u305f\u3081\u306b\u306f\uff0c\uff08$y^{”}$ \u306e\u300c\u4fc2\u6570\u300d\u306f\u3069\u3061\u3089\u3082 $f(x)$ \u3067\u7b49\u3057\u3044\u306e\u3067\uff09$y’$ \u306e\u300c\u4fc2\u6570\u300d\u3068 $y$ \u306e\u300c\u4fc2\u6570\u300d\u304c\u7b49\u3057\u304f\u306a\u308b\u5fc5\u8981\u304c\u3042\u308a\uff0c<\/p>\n
\\begin{eqnarray}
\nf'(x) &=& b f(x) \\\\
\nf^{”}(x) + K f(x)&=& c f(x)\\\\
\n\\therefore\\ \\\u00a0 f(x) &=& e^{b x} \\\\
\n\\therefore\\ \\\u00a0 K &=& c-b^2
\n\\end{eqnarray}<\/p>\n
\u3053\u306e $f(x)$ \u3092\u4f7f\u3046\u3068\u5143\u306e\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f<\/p>\n
$$\\bigl\\{f(x) y \\bigr\\}^{”} = -K \\bigl\\{f(x) y \\bigr\\}$$<\/p>\n
\u3068\u306a\u308b\u306e\u3060\u304b\u3089\uff0c$K > 0$ \u3059\u306a\u308f\u3061 $c\u00a0 -b^2 > 0$ \u306e\u3068\u304d\u306b\u306f\uff0c\u305f\u3060\u3061\u306b\uff08\u810a\u9ac4\u53cd\u5c04\u7684\u306b\uff09<\/p>\n
$$ \\bigl\\{f(x) y \\bigr\\} = A \\cos \\bigl(\\sqrt{c\u00a0 -b^2} \\,x\\bigr) + B \\sin \\bigl(\\sqrt{c\u00a0 -b^2} \\,x\\bigr)$$<\/p>\n
$f(x) = e^{bx}$ \u3092\u4ee3\u5165\u3057\u3066 $y$ \u306b\u3064\u3044\u3066\u66f8\u304d\u51fa\u3059\u3068\uff0c<\/p>\n
$$y = e^{-b x} \\left\\{A \\cos \\bigl(\\sqrt{c\u00a0 -b^2} \\,x\\bigr) + B\u00a0 \\sin \\bigl(\\sqrt{c\u00a0 -b^2} \\,x \\bigr)\\right\\} $$<\/p>\n
$c -b^2 > 0$ \u4ee5\u5916\u306e\u5834\u5408\u306b\u3082\u9069\u7528\u3067\u304d\u308b\u3088\u3046\u306b\uff0c\u3042\u3089\u305f\u3081\u3066\u7a4d\u5206\u5b9a\u6570\u3092 $\\displaystyle B \\rightarrow \\frac{B}{\\sqrt{c-b^2}}$ \u3068\u7f6e\u304d\u76f4\u3057\u3066\u66f8\u304f\u3068\uff0c<\/p>\n
$$y = e^{-b x} \\left\\{A \\cos \\bigl(\\sqrt{c -b^2} \\,x\\bigr) + \\frac{B}{\\sqrt{c -b^2} } \\sin \\bigl(\\sqrt{c -b^2} \\,x \\bigr)\\right\\} $$<\/p>\n
$c -b^2 < 0$ \u306e\u5834\u5408\u3084 $c -b^2 = 0$\u00a0 \u306e\u5834\u5408\u3082\uff0c\u3053\u306e\u5f0f\u304b\u3089\u305f\u3060\u3061\u306b\u5f97\u3089\u308c\u308b\u3002<\/p>\n
\\(c -b^2\u00a0 < 0\\) \u306e\u5834\u5408\u306f\uff0c$\\sqrt{c-b^2} = i \\sqrt{b^2-c}$ \u3092\u4f7f\u3063\u3066
\n\\begin{eqnarray}
\ny &=& e^{-b x} \\left\\{ A \\cos \\bigl(i \\sqrt{b^2-c} \\,x\\bigr) + \\frac{B}{i \\sqrt{b^2-c}} \\sin \\bigl(i \\sqrt{b^2-c} \\,x\\bigr) \\right\\}\\\\
\n&=& e^{-b x} \\left\\{ A \\cosh \\bigl(\\sqrt{b^2-c} \\,x\\bigr) + \\frac{B}{\\sqrt{b^2-c}} \\sinh \\bigl(\\sqrt{b^2-c} \\,x\\bigr) \\right\\}
\n\\end{eqnarray}<\/p>\n
\u3044\u3063\u305f\u3093\u3053\u306e\u5f62\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u3089\uff0c\u3042\u3089\u305f\u3081\u3066 $\\displaystyle \\frac{B}{\\sqrt{b^2-c}} \\rightarrow B$\u00a0 \u3068\u7f6e\u304d\u623b\u3057\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3044\u3066\u3082\u3088\u3044\u3067\u3042\u308d\u3046\u3002<\/p>\n
$$y = e^{-b x} \\left\\{ A \\cosh \\bigl(\\sqrt{b^2-c} \\,x\\bigr) + B\u00a0 \\sinh \\bigl(\\sqrt{b^2-c} \\,x\\bigr) \\right\\} $$<\/p>\n
\u3042\u308b\u3044\u306f\uff0c$y^{”} = y$ \u306e\u7b54\u3048\u304c\u810a\u9ac4\u53cd\u5c04\u3067\u300c2\u3064\u306e\u89e3\u306f \\(e^x\\) \u3068\\(e^{-x}\\) \u3060\u30fc\u3063<\/strong><\/span>\u300d\u3068\u53eb\u3079\u308b\u306e\u3067\u3042\u308c\u3070\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3044\u3066\u3082\u3088\u3044\u3002<\/p>\n$$y = e^{-b x} \\left\\{ C_1\\exp \\bigl(\\sqrt{b^2-c} \\,x\\bigr) + C_2\u00a0 \\exp \\bigl(-\\sqrt{b^2-c} \\,x\\bigr) \\right\\} $$<\/p>\n
\\(c -b^2 = 0\\) \u306e\u5834\u5408\u306f\uff0c\\(c -b^2 \\rightarrow 0\\) \u306e\u6975\u9650\u3092\u3068\u3063\u3066
\n\\begin{eqnarray}
\ny
\n&=& \\lim_{(c-b^2)\u00a0 \\rightarrow 0} e^{-b x} \\left\\{A \\cos \\bigl(\\sqrt{c -b^2} \\,x\\bigr) + \\frac{B}{\\sqrt{c -b^2} } \\sin \\bigl(\\sqrt{c -b^2} \\,x \\bigr)\\right\\} \\\\
\n&=& e^{-b x} \\left\\{ A + {B} \\,x\\right\\}
\n\\end{eqnarray}<\/p>\n
\u307e\u3068\u3081<\/h4>\n
\\( y^{”} + 2 b\\, y’ + c\\, y = 0 \\) \u306f\uff0c\\(Y(x) \\equiv e^{b x}\\,\u00a0 y\\) \u306b\u5bfe\u3057\u3066\u6700\u3082\u7c21\u5358\u306a\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f<\/p>\n
$$Y^{\\prime\\prime} + K\\, Y = 0, \\quad K \\equiv c-b^2$$<\/p>\n
\u306b\u306a\u308b\u306e\u3067\uff0c\u810a\u9ac4\u53cd\u5c04\u3067\u305f\u3060\u3061\u306b<\/p>\n
$$ Y =e^{b x}\\,\u00a0 y = \\left\\{
\n\\begin{array}{ll}
\nA \\cos \\sqrt{K}\\ x + B \\sin \\sqrt{K} \\ x & (K > 0)\\\\ \\ \\\\
\nA \\cosh \\sqrt{-K} \\ x + B \\sinh \\sqrt{-K} \\ x & (K < 0) \\\\ \\ \\\\
\nA + B x & (K = 0)
\n\\end{array}
\n\\right.$$<\/p>\n
\u3059\u306a\u308f\u3061\uff0c<\/p>\n
$$ y = \\left\\{
\n\\begin{array}{ll}
\ne^{-b x} \\left( A \\cos \\sqrt{c-b^2} \\ x + B \\sin \\sqrt{c-b^2} \\ x \\right)& (c-b^2 > 0)\\\\ \\ \\\\
\ne^{-b x} \\left( A \\cosh \\sqrt{b^2 -c} \\ x + B \\sinh \\sqrt{b^2 -c} \\ x \\right)& (c-b^2 < 0) \\\\ \\ \\\\
\ne^{-b x} \\left( A + B x\\right) & (c-b^2 = 0)
\n\\end{array}
\n\\right.$$<\/p>\n
$c-b^2 < 0$ \u306e\u5834\u5408\u306b\u3064\u3044\u3066\u306f\uff0c\u53cc\u66f2\u7dda\u95a2\u6570\u3092\u4f7f\u308f\u305a\u306b\u6307\u6570\u95a2\u6570\u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u3082\u3067\u304d\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n
$$y = C_1 \\exp\\biggl(\\left(-b + \\sqrt{b^2 -c}\\right) x\\biggr) + C_2 \\exp\\biggl(\\left(-b -\\sqrt{b^2 -c}\\right) x\\biggr)$$<\/p>\n
\u53c2\u8003\uff1a\u4e00\u822c\u7684\u306a\u6559\u79d1\u66f8\u306e\u8aac\u660e\u4f8b<\/h3>\n
\u4e00\u822c\u7684\u306a\u6559\u79d1\u66f8\u3067\u306f\uff0c\u89e3\u3092 \\( y = e^{\\lambda x} \\) \uff08\\(\\lambda\\) \u306f\u30ae\u30ea\u30b7\u30a2\u6587\u5b57\u306e\u30e9\u30e0\u30c0\u306e\u5c0f\u6587\u5b57\u3067\uff0c\u3053\u3053\u3067\u306f\u5b9a\u6570\u3092\u3042\u3089\u308f\u3059\uff09\u306e\u5f62\u306b\u6c7a\u3081\u3046\u3061\u3057\u3066\uff08\u306a\u305c\u305d\u3046\u4eee\u5b9a\u3059\u308b\u306e\u304b\u3063\u3066\uff1f \u7d9a\u304d\u3092\u8aad\u3081\u3070\u308f\u304b\u308b\u3088\u3046\u306b\uff0c\u3053\u3046\u4eee\u5b9a\u3059\u308b\u3068\u89e3\u3051\u308b\u304b\u3089\u3067\u3059\u3088\uff09\uff0c\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u4ee3\u5165\u3057\uff0c\u5b9a\u6570 $\\lambda$ \u306b\u3064\u3044\u3066\u306e2\u6b21\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3068\u3044\u3046\u7acb\u5834\u3092\u3068\u308b\u3088\u3046\u3060\u3002<\/p>\n
$$y^{”} + 2 b\\, y’ + c\\, y = 0 $$<\/p>\n
\u3059\u3067\u306b\u89e3\u3092 \\( y = e^{\\lambda x} \\) \u306e\u5f62\u306b\u4eee\u5b9a\u3057\u3066\u4ee3\u5165\u3059\u308b\u3068\u89e3\u3051\u308b\u3068\u3044\u3046\u524d\u4f8b\u304c\u3042\u3063\u305f\u306e\u3067\uff0c\u3053\u3053\u3067\u3082\u305d\u3046\u3059\u308b\u3002
\n$$y^{”} + 2 b\\, y’ + c\\, y = \\left( \\lambda^2 + 2 b \\lambda + c \\right) e^{\\lambda x}= 0 $$<\/p>\n
\u3053\u308c\u3088\u308a\uff0c\\(\\lambda\\) \u306f2\u6b21\u65b9\u7a0b\u5f0f \\( \\lambda^2 + 2 b \\lambda + c = 0 \\) \u306e\u89e3\u3002<\/p>\n
\\( b^2 -c > 0 \\) \u306e\u5834\u5408<\/h4>\n
2\u3064\u306e\u5b9f\u6570\u89e3\u304c\u5b58\u5728\u3057\uff0c
\n$$ \\lambda = -b \\pm \\sqrt{b^2 -c}$$
\n\\( \\mu \\equiv \\sqrt{b^2 -c} \\) \u3068\u5b9a\u7fa9\u3057\u3066\u304a\u304f\u3068\uff0c$$ \\lambda_1 = -b + \\mu, \\quad \\lambda_2 = -b -\\mu $$<\/p>\n
\u4e00\u822c\u89e3\u306f2\u3064\u306e1\u6b21\u72ec\u7acb\u306a\u89e3\u306e\u7dda\u5f62\u548c\u3067\u3042\u308b\u304b\u3089
\n\\begin{eqnarray}
\ny &=& C_1 e^{\\lambda_1\\, x} + C_2 e^{\\lambda_2\\, x} \\\\
\n&=& e^{-b x} \\biggl\\{ C_1 e^{\\mu x} + C_2 e^{-\\mu x} \\biggr\\} \\\\
\n&=& e^{-b x} \\biggl\\{ C_1 \\bigl( \\cosh \\left(\\mu x\\right) + \\sinh \\left(\\mu x\\right) \\bigr)\u00a0 + C_2 \\bigl( \\cosh \\left(\\mu x\\right) -\\sinh \\left(\\mu x\\right) \\bigr)\u00a0 \\biggr\\}\\\\
\n&=& e^{-b x} \\biggl\\{ A \\cosh \\left(\\mu x\\right) + B \\sinh \\left(\\mu x\\right) \\biggr\\}
\n\\end{eqnarray}
\n\u3053\u3053\u3067\uff0c\\( A = C_1 + C_2, \\ B = C_1 -C_2 \\) \u3068\u304a\u3044\u305f\u3002<\/p>\n
<\/p>\n
\\( b^2 -c < 0 \\) \u306e\u5834\u5408<\/h4>\n
2\u3064\u306e\u8907\u7d20\u5171\u5f79\u89e3\u304c\u5b58\u5728\u3057\uff0c
\n$$ \\lambda = -b \\pm i \\sqrt{c -b^2}$$
\n\\( \\nu \\equiv \\sqrt{c -b^2} \\) \u3068\u5b9a\u7fa9\u3057\u3066\u304a\u304f\u3068\uff0c$$ \\lambda_1 = -b + i \\nu, \\quad \\lambda_2 = -b -i \\nu $$<\/p>\n
<\/p>\n
\u4e00\u822c\u89e3\u306f
\n\\begin{eqnarray}
\ny &=& C_1 e^{\\lambda_1 x} + C_2 e^{\\lambda_2 x} \\\\
\n&=& e^{-b x} \\left\\{ C_1 e^{i \\nu\\, x} + C_2 e^{-i\\nu\\, x} \\right\\} \\\\
\n&=& e^{-b x} \\biggl\\{ C_1 \\bigl( \\cos (\\nu x) + i \\sin (\\nu x) \\bigr)\u00a0 + C_2 \\bigl( \\cos (\\nu x) -i\\sin (\\nu x) \\bigr)\u00a0 \\biggr\\}\\\\
\n&=& e^{-b x} \\biggl\\{ A \\cos (\\nu x) + B \\sin (\\nu x) \\biggr\\}
\n\\end{eqnarray}<\/p>\n
\u3053\u3053\u3067\uff0c\\( A = C_1 + C_2, \\ B = i \\left(C_1 -C_2\\right) \\) \u3068\u304a\u3044\u305f\u3002<\/p>\n
<\/p>\n
\\( b^2 -c = 0 \\) \u306e\u5834\u5408<\/h4>\n
1\u3064\u306e\u91cd\u89e3\u304c\u5b58\u5728\u3057\uff0c$$ \\lambda = -b $$ \u5f93\u3063\u3066\u4e00\u822c\u89e3\u306f $$ y = C e^{\\lambda x} = C e^{-b x}$$<\/p>\n
<\/p>\n
… \u3068\u3057\u3066\u307b\u3063\u3068\u3057\u3066\u3044\u3066\u306f\u3044\u3051\u306a\u3044<\/b><\/u><\/span>\u3002\u306a\u305c\u304b\uff1f<\/p>\n\u305d\u308c\u306f\uff0c\u300c2\u968e\u7dda\u5f62\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u306f2\u3064\u306e1\u6b21\u72ec\u7acb\u306a\u89e3<\/b><\/u><\/span>\u304c\u3042\u308b\u306f\u305a\u3060\u304b\u3089\u3002\u300d<\/p>\n\u305d\u3053\u3067\uff0c\u3053\u3053\u3067\u306f\u4f8b\u5916\u7684\u306b\uff0c\u7a4d\u5206\u5b9a\u6570 \\( C \\) \u3092\u5909\u6570 \\( x \\) \u306b\u4f9d\u5b58\u3059\u308b\uff0c\u3064\u307e\u308a \\( C(x) \\) \u3068\u601d\u3044\u76f4\u3057\u3066\u89e3\u3044\u3066\u307f\u308b\u3002\uff08\u5b9a\u6570\u5909\u5316\u6cd5<\/b><\/span>\u3068\u304b\u4fc2\u6570\u5909\u5316\u6cd5<\/b><\/span>\u3068\u304b\u547c\u3070\u308c\u308b\u624b\u6cd5\u3067\u3042\u308b\u3002\uff09<\/p>\n\\( y = C(x) e^{-b x} \\) \u3068\u304a\u304f\u3068\uff0c\\( y’ = (C’ -b C) e^{-b x}, \\ y^{”} = (C^{”} -2b C’ + b^2 C)\u00a0 e^{-b x} \\) \u3068\u306a\u308a\uff0c
\n$$ y^{”} + 2 b\\, y’ + c\\, y = C^{”} e^{-b x} = 0, \\quad \\therefore C^{”} = 0, \\quad \\therefore C = A + B x$$
\n\u3068\u306a\u3063\u3066\u4e00\u822c\u89e3\u306f\uff0c$$ y = e^{-b x} \\left\\{ A + B x\\right\\} $$<\/p>\n
\u307e\u3068\u3081<\/h4>\n
\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u540c\u6b21\u65b9\u7a0b\u5f0f \\( y^{”} + 2 b\\, y’ + c\\, y = 0 \\) \u306e\u4e00\u822c\u89e3\u306f…<\/p>\n
<\/p>\n
\\( b^2\u00a0 -c > 0 \\) \u306e\u5834\u5408<\/h5>\n
$$
\ny = e^{-b x} \\biggl\\{ A \\cosh (\\mu x) + B \\sinh (\\mu x) \\biggr\\}, \\quad \\mu \\equiv \\sqrt{b^2\u00a0 -c}
\n$$<\/p>\n
\\( b^2 -c < 0 \\) \u306e\u5834\u5408<\/h5>\n
$$
\ny = e^{-b x} \\biggl\\{ A \\cos (\\nu x) + B \\sin (\\nu x) \\biggr\\}, \\quad \\nu \\equiv \\sqrt{c\u00a0 -b^2}
\n$$<\/p>\n
\\( b^2\u00a0 -c = 0 \\) \u306e\u5834\u5408<\/h5>\n
$$ y = e^{-b x} \\bigl\\{ A + B x\\bigr\\} $$<\/p>\n
\u3069\u3046\u3067\u3059\u304b\uff1f \u8aac\u660e\u304c\u9577\u304b\u3063\u305f\u3051\u3069\uff0c\u4e00\u822c\u89e3\u304c\u8868\u73fe\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u540c\u6b21\u65b9\u7a0b\u5f0f \\( y^{”} + 2 b\\, y’ + c\\, y = 0 \\) \u306e\u4e00\u822c\u89e3\u3092\u6c42\u3081\u308b\u3002\u3067\u304d\u308c\u3070\u7d71\u4e00\u7684\u306a\u8868\u8a18\u3067\u3002<\/p>
\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":2224,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2272","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\/2272","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=2272"}],"version-history":[{"count":26,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2272\/revisions"}],"predecessor-version":[{"id":10231,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2272\/revisions\/10231"}],"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=2272"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}