{"id":2265,"date":"2022-02-24T18:36:18","date_gmt":"2022-02-24T09:36:18","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2265"},"modified":"2025-05-05T12:56:28","modified_gmt":"2025-05-05T03:56:28","slug":"%e6%9c%80%e3%82%82%e7%b0%a1%e5%8d%98%e3%81%aa%e5%ae%9a%e6%95%b0%e4%bf%82%e6%95%b02%e9%9a%8e%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%bc%8f%ef%bc%9a%e7%b6%9a%e3%81%8d","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\/%e6%9c%80%e3%82%82%e7%b0%a1%e5%8d%98%e3%81%aa%e5%ae%9a%e6%95%b0%e4%bf%82%e6%95%b02%e9%9a%8e%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%e5%bc%8f%ef%bc%9a%e7%b6%9a%e3%81%8d\/","title":{"rendered":"\u6700\u3082\u7c21\u5358\u306a\u5b9a\u6570\u4fc2\u65702\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\uff1a\u7d9a\u304d"},"content":{"rendered":"

\\( y^{”} +\u00a0 K y = 0\\) \u3042\u308b\u3044\u306f\u79fb\u9805\u3057\u3066 \\(y^{”} = – K y\\) \u3092\u89e3\u304f\u3002
\n<\/p>\n

\\(y^{”} = – K y \\ (K > 0)\\) \u3092\u810a\u9ac4\u53cd\u5c04\u3067\u89e3\u304f<\/h3>\n

$y^{”} = -y$ \u306e\u7b54\u3048\u304c\u810a\u9ac4\u53cd\u5c04\u3067\u300c2\u3064\u306e\u89e3\u306f $\\cos x$ \u3068 $\\sin x$ \u3060\u30fc\u3063<\/strong><\/span>\u300d\u3068\u53eb\u3079\u308b\u306e\u3067\u3042\u308c\u3070\uff0c$K>0$ \u306e\u5834\u5408\u306e $y^{”} =- K y$ \u306e\u7b54\u3048\u3082\u810a\u9ac4\u53cd\u5c04\u3067\u308f\u304b\u308b\u306f\u305a\u3002<\/p>\n

\\begin{eqnarray}
\ny^{”} = \\frac{d^2 y}{dx^2} &=& – K y \\\\
\n\\frac{d^2 y}{K dx^2} &=& – y \\\\
\n&& (\\tilde{x} \\equiv \\sqrt{K} x ) \\\\
\n\\frac{d^2 y}{d\\tilde{x}^2} &=& – y \\\\
\n\\therefore\\ \\ y &=& C_1 \\cos \\tilde{x} + C_2 \\sin \\tilde{x} \\\\
\n&=& C_1 \\cos \\sqrt{K} x + C_2 \\sin \\sqrt{K} x
\n\\end{eqnarray}<\/p>\n

\\(y^{”} = – K y \\ (K < 0) \\) \u3092\u810a\u9ac4\u53cd\u5c04\u3067\u89e3\u304f<\/h3>\n

$y^{”} = y$ \u306e\u7b54\u3048\u304c\u810a\u9ac4\u53cd\u5c04\u3067\u300c2\u3064\u306e\u89e3\u306f\u53cc\u66f2\u7dda\u95a2\u6570\u306e $\\cosh x$ \u3068 $\\sinh x$ \u3060\u30fc\u3063<\/strong><\/span>\u300d\u3068\u53eb\u3079\u308b\u306e\u3067\u3042\u308c\u3070\uff0c$K<0$ \u306e\u5834\u5408\u306e $y^{”} =- K y = |K| y$ \u306e\u7b54\u3048\u3082\u810a\u9ac4\u53cd\u5c04\u3067\u308f\u304b\u308b\u306f\u305a\u3002<\/p>\n

\\begin{eqnarray}
\ny^{”} = \\frac{d^2 y}{dx^2} &=& |K| y \\\\
\n\\frac{d^2 y}{|K| dx^2} &=&\u00a0 y \\\\
\n&& (\\tilde{x} \\equiv \\sqrt{|K|} x ) \\\\
\n\\frac{d^2 y}{d\\tilde{x}^2} &=&\u00a0 y \\\\
\n\\therefore\\ \\ y &=& C_1 \\cosh \\tilde{x} + C_2 \\sinh \\tilde{x} \\\\
\n&=& C_1 \\cosh \\sqrt{|K|} x + C_2 \\sinh \\sqrt{|K|} x
\n\\end{eqnarray}<\/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

\\begin{eqnarray}
\ny &=& C_1 \\exp\\left( \\sqrt{|K|} x \\right) + C_2 \\exp\\left( -\\sqrt{|K|} x \\right)
\n\\end{eqnarray}<\/p>\n

\\(y^{”} = 0 \\ (K = 0)\\) \u306f…<\/h3>\n

2\u968e\u5fae\u5206\u304c\u30bc\u30ed\u3067\u3042\u308b\u304b\u3089\u305f\u3060\u3061\u306b2\u3064\u306e\u7a4d\u5206\u5b9a\u6570 $C_1, \\, C_2$ \u3092\u4f7f\u3063\u3066<\/p>\n

$$y(x) = C_1 + C_2 x$$<\/p>\n

\u7d71\u4e00\u7684\u306a\u8868\u8a18\u6cd5\u3067…<\/h3>\n

$K>0$ \u306e\u6642\u306e\u4e00\u822c\u89e3\u306e\u7a4d\u5206\u5b9a\u6570\u3092\u3042\u3089\u305f\u3081\u3066 $C_1 \\rightarrow A$, $\\displaystyle C_2 \\rightarrow \\frac{B}{\\sqrt{K}}$ \u3068\u304a\u3044\u3066\u3084\u308b\u3068<\/p>\n

$$y(x) = A \\cos \\sqrt{K} x + \\frac{B}{\\sqrt{K}} \\sin \\sqrt{K} x$$<\/p>\n

$K < 0$ \u306e\u3068\u304d\u3082\u305d\u306e\u307e\u307e\uff0c$K\u00a0 = – |K| $ \u3092\u4ee3\u5165\u3057\u3066<\/p>\n

\\begin{eqnarray}
\ny(x) &=& A \\cos \\sqrt{- |K| } x + \\frac{B}{\\sqrt{- |K| } } \\sin \\sqrt{- |K| } x \\\\
\n&=& A \\cos(i \\sqrt{|K|} x)\u00a0 + \\frac{B}{i \\sqrt{|K|} } \\sin (i \\sqrt{|K|} x) \\\\
\n&=& A \\cosh(\\sqrt{|K|} x) + \\frac{B}{\\sqrt{|K|} } \\sinh (\\sqrt{|K|} x)
\n\\end{eqnarray}<\/p>\n

\u3060\u3057\uff0c$K=0$ \u306e\u5834\u5408\u306f
\n\\begin{eqnarray}
\ny &=& \\lim_{K\\rightarrow 0} \\left(A \\cos \\sqrt{K} x + \\frac{B}{\\sqrt{K} } \\sin \\sqrt{K} x \\right) \\\\
\n&=& A + B x
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308a\uff0c$K > 0$ \u306e\u3068\u304d\u306e\u89e3\u3067 $K < 0$ \u306e\u5834\u5408\u3082 $K=0$ \u306e\u5834\u5408\u3082\u7d71\u4e00\u7684\u306b\u3042\u3089\u308f\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3053\u308c\u304c\uff0c\u4eba\u751f\u306b\u53cc\u66f2\u7dda\u95a2\u6570\u304c\u5fc5\u8981\u3067\u3042\u308b\u7406\u7531\uff01<\/p>\n

\u307e\u3068\u3081<\/h3>\n

\\( y^{”} +K y = 0\\) \u3042\u308b\u3044\u306f\u79fb\u9805\u3057\u3066 \\( y^{”} = -K y\\) \u306e\u89e3\u306f\uff0c\uff08\u7a4d\u5206\u5b9a\u6570\u3092 $A, B$ \u306b\u7f6e\u304d\u623b\u3057\u3066\uff09<\/p>\n

\\begin{eqnarray}
\nK > 0 &\\ \\ \\Longrightarrow \\ \\\u00a0 & y = A \\cos(\\sqrt{K} \\ x) + B \\sin(\\sqrt{K} \\ x)\\\\
\nK < 0 &\\ \\ \\Longrightarrow \\ \\\u00a0 & y = A \\cosh(\\sqrt{|K|} \\ x) + B \\sinh(\\sqrt{|K|} \\ x)\\\\
\nK = 0 &\\ \\ \\Longrightarrow \\ \\\u00a0 & y = A\u00a0 + B\\ x
\n\\end{eqnarray}<\/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\u89e3\u304f\u3068\u3044\u3046\u7acb\u5834\u3092\u3068\u308b\u3088\u3046\u3060\u3002<\/p>\n

\\( K>0 \\) \u306e\u5834\u5408<\/h4>\n

\u89e3\u3092 \\( y = e^{\\lambda x} \\) \u306e\u5f62\u306b\u4eee\u5b9a\u3057\u3066\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u5165\u308c\u308b\u3068\uff0c<\/p>\n

$$ \\frac{d^2}{dx^2} e^{\\lambda x} = \\lambda^2 e^{\\lambda x} = -K e^{\\lambda x} $$<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0c\\( \\lambda = \\pm i\\sqrt{K} \\) \u3068\u306a\u308a\uff0c\u4e00\u822c\u89e3\u306f2\u3064\u306e\u4e00\u6b21\u72ec\u7acb\u306a\u89e3\u306e\u7dda\u5f62\u548c\u3067\uff0c<\/p>\n

$$ y = C_1 e^{+i\\sqrt{K} x} + C_2 e^{-i\\sqrt{K} x} $$<\/p>\n

…\u3068\u306a\u308b\u3051\u3069\uff0c\u3067\u3082\uff0c\u6307\u6570\u95a2\u6570\u306e\u65b9\u306b\u865a\u6570\u304c\u4e57\u308b\u3063\u3066\u3069\u3046\u3044\u3046\u3053\u3068!? \u3068\u9996\u3092\u304b\u3057\u3052\u308b\u3042\u306a\u305f\u306e\u305f\u3081\u306b\uff0c\u300c\u4eba\u985e\u306e\u81f3\u5b9d\uff1a\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u300d\u3068\u3044\u3046\u30da\u30fc\u30b8\u3092\u5225\u306b\u4f5c\u308a\u307e\u3057\u305f\u3002\u8a73\u7d30\u306f\u305d\u3061\u3089\u3092\u307f\u3066\u3044\u305f\u3060\u304f\u3068\u3057\u3066\uff0c\u3053\u306e\u30aa\u30a4\u30e9\u30fc\u306e\u516c\u5f0f\u3092\u4f7f\u3046\u3068\uff0c
\n\\begin{eqnarray}
\ne^{+i \\sqrt{K} x} &=& \\cos(\\sqrt{K} x) + i \\sin(\\sqrt{K} x) \\\\
\ne^{-i \\sqrt{K} x} &=& \\cos(\\sqrt{K} x) – i \\sin(\\sqrt{K} x)
\n\\end{eqnarray}<\/p>\n

\u3067\u3042\u308b\u304b\u3089
\n\\begin{eqnarray}
\ny &=& C_1 e^{+i \\sqrt{K} x} + C_2 e^{-i\\sqrt{K} x} \\\\
\n&=& C_1 \\left( \\cos(\\sqrt{K} x) + i \\sin(\\sqrt{K} x)\\right) + C_2\\left( \\cos(\\sqrt{K} x) – i \\sin(\\sqrt{K} x)\\right)\\\\
\n&=& \\left( C_1 + C_2\\right) \\cos(\\sqrt{K} x) + i \\left( C_1 – C_2 \\right)\\sin(\\sqrt{K} x)
\n\\end{eqnarray}<\/p>\n

 <\/p>\n

\u3053\u3053\u3067\u3042\u3089\u305f\u3081\u3066 \\( C_1 + C_2 \\Rightarrow C’_1, \\ \\ i \\left( C_1 – C_2 \\right) \\Rightarrow C’_2\\) \u3068\u7f6e\u304d\u76f4\u3059\u3068
\n$$y = C’_1 \\cos(\\sqrt{K} x) + C’_2 \\sin(\\sqrt{K} x)$$\u00a0 \u3068\u306a\u308a\uff0c\u3053\u308c\u306f\uff0c\u3053\u306e\u624b\u306e2\u968e\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u810a\u9ac4\u53cd\u5c04\u3067\u89e3\u3044\u305f\u3068\u304d\u306e\u7b54\u3048\u3068\u4e00\u81f4\u3059\u308b\u3002<\/p>\n

\\( K = 0 \\) \u306e\u5834\u5408<\/h4>\n

\\( y^{”} = 0 \\) \u3068\u306a\u308b\u306e\u3067\uff0c\u7c21\u5358\u306b\u89e3\u3051\u3066 \\( y = C_1\u00a0 + C_2\\, x\\)\u3002<\/p>\n

\\( K < 0 \\) \u306e\u5834\u5408<\/h4>\n

\u3053\u306e\u3068\u304d\u306f\uff0c\\( K = – |K|\\) \u3068\u306a\u308b\u304b\u3089\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f<\/p>\n

$$ y^{”} = |K| y$$<\/p>\n

\u3053\u3053\u3067\u3082\u89e3\u3092 \\( y = e^{\\lambda x} \\) \u306e\u5f62\u306b\u4eee\u5b9a\u3057\u3066\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u3044\u308c\u308b\u3068\uff0c<\/p>\n

$$ \\frac{d^2}{dx^2} e^{\\lambda x} =\\lambda^2 e^{\\lambda x} =\u00a0 |K| e^{\\lambda x} $$<\/p>\n

\u6a5f\u68b0\u7684\u306b\u89e3\u304f\u3068 \\( \\lambda = \\pm\u00a0 \\sqrt{|K|} \\) \u3068\u306a\u308a\uff0c\u4e00\u822c\u89e3\u306f2\u3064\u306e\u4e00\u6b21\u72ec\u7acb\u306a\u89e3\u306e\u7dda\u5f62\u548c\u3067\uff0c<\/p>\n

$$ y = C_1 e^{+ \\sqrt{|K|} x} + C_2 e^{-\\sqrt{|K|} x} $$<\/p>\n

 <\/p>\n

\u307e\u3068\u3081<\/h4>\n

\\( y^{”} +K y = 0\\) \u3042\u308b\u3044\u306f\u79fb\u9805\u3057\u3066 \\( y^{”} = -K y\\) \u306e\u89e3\u306f\uff0c\uff08\u7a4d\u5206\u5b9a\u6570\u3092\u3042\u3089\u305f\u3081\u3066 $A, \\, B$ \u3068\u3057\u3066\uff09
\n\\begin{eqnarray}
\nK > 0 &\\ \\ \\Longrightarrow \\ \\\u00a0 & y = A \\cos(\\sqrt{K} \\ x) + B \\sin(\\sqrt{K} \\ x)\\\\
\nK < 0 &\\ \\ \\Longrightarrow \\ \\\u00a0 & y = A e^{+\\sqrt{|K|} \\ x} + B e^{-\\sqrt{|K|}\\\u00a0 x}\\\\
\nK = 0 &\\ \\ \\Longrightarrow \\ \\\u00a0 & y = A\u00a0 + B x
\n\\end{eqnarray}<\/p>\n","protected":false},"excerpt":{"rendered":"

\\( y^{”} +\u00a0 K y = 0\\) \u3042\u308b\u3044\u306f\u79fb\u9805\u3057\u3066 \\(y^{”} = – K y\\) \u3092\u89e3\u304f\u3002<\/p>

\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":2224,"menu_order":8,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2265","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\/2265","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=2265"}],"version-history":[{"count":17,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2265\/revisions"}],"predecessor-version":[{"id":10225,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2265\/revisions\/10225"}],"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=2265"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}