{"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":"<p>\\( y^{&#8221;} +\u00a0 K y = 0\\) \u3042\u308b\u3044\u306f\u79fb\u9805\u3057\u3066 \\(y^{&#8221;} = &#8211; K y\\) \u3092\u89e3\u304f\u3002<br \/>\n<!--more--><\/p>\n<h3>\\(y^{&#8221;} = &#8211; K y \\ (K &gt; 0)\\) \u3092\u810a\u9ac4\u53cd\u5c04\u3067\u89e3\u304f<\/h3>\n<p>$y^{&#8221;} = -y$ \u306e\u7b54\u3048\u304c\u810a\u9ac4\u53cd\u5c04\u3067\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>2\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$K&gt;0$ \u306e\u5834\u5408\u306e $y^{&#8221;} =- K y$ \u306e\u7b54\u3048\u3082\u810a\u9ac4\u53cd\u5c04\u3067\u308f\u304b\u308b\u306f\u305a\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\ny^{&#8221;} = \\frac{d^2 y}{dx^2} &amp;=&amp; &#8211; K y \\\\<br \/>\n\\frac{d^2 y}{K dx^2} &amp;=&amp; &#8211; y \\\\<br \/>\n&amp;&amp; (\\tilde{x} \\equiv \\sqrt{K} x ) \\\\<br \/>\n\\frac{d^2 y}{d\\tilde{x}^2} &amp;=&amp; &#8211; y \\\\<br \/>\n\\therefore\\ \\ y &amp;=&amp; C_1 \\cos \\tilde{x} + C_2 \\sin \\tilde{x} \\\\<br \/>\n&amp;=&amp; C_1 \\cos \\sqrt{K} x + C_2 \\sin \\sqrt{K} x<br \/>\n\\end{eqnarray}<\/p>\n<h3>\\(y^{&#8221;} = &#8211; K y \\ (K &lt; 0) \\) \u3092\u810a\u9ac4\u53cd\u5c04\u3067\u89e3\u304f<\/h3>\n<p>$y^{&#8221;} = y$ \u306e\u7b54\u3048\u304c\u810a\u9ac4\u53cd\u5c04\u3067\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>2\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$K&lt;0$ \u306e\u5834\u5408\u306e $y^{&#8221;} =- K y = |K| y$ \u306e\u7b54\u3048\u3082\u810a\u9ac4\u53cd\u5c04\u3067\u308f\u304b\u308b\u306f\u305a\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\ny^{&#8221;} = \\frac{d^2 y}{dx^2} &amp;=&amp; |K| y \\\\<br \/>\n\\frac{d^2 y}{|K| dx^2} &amp;=&amp;\u00a0 y \\\\<br \/>\n&amp;&amp; (\\tilde{x} \\equiv \\sqrt{|K|} x ) \\\\<br \/>\n\\frac{d^2 y}{d\\tilde{x}^2} &amp;=&amp;\u00a0 y \\\\<br \/>\n\\therefore\\ \\ y &amp;=&amp; C_1 \\cosh \\tilde{x} + C_2 \\sinh \\tilde{x} \\\\<br \/>\n&amp;=&amp; C_1 \\cosh \\sqrt{|K|} x + C_2 \\sinh \\sqrt{|K|} x<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3042\u308b\u3044\u306f\uff0c$y^{&#8221;} = y$ \u306e\u7b54\u3048\u304c\u810a\u9ac4\u53cd\u5c04\u3067\u300c<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>2\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<p>\\begin{eqnarray}<br \/>\ny &amp;=&amp; C_1 \\exp\\left( \\sqrt{|K|} x \\right) + C_2 \\exp\\left( -\\sqrt{|K|} x \\right)<br \/>\n\\end{eqnarray}<\/p>\n<h3>\\(y^{&#8221;} = 0 \\ (K = 0)\\) \u306f&#8230;<\/h3>\n<p>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<p>$$y(x) = C_1 + C_2 x$$<\/p>\n<h3>\u7d71\u4e00\u7684\u306a\u8868\u8a18\u6cd5\u3067&#8230;<\/h3>\n<p>$K&gt;0$ \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<p>$$y(x) = A \\cos \\sqrt{K} x + \\frac{B}{\\sqrt{K}} \\sin \\sqrt{K} x$$<\/p>\n<p>$K &lt; 0$ \u306e\u3068\u304d\u3082\u305d\u306e\u307e\u307e\uff0c$K\u00a0 = &#8211; |K| $ \u3092\u4ee3\u5165\u3057\u3066<\/p>\n<p>\\begin{eqnarray}<br \/>\ny(x) &amp;=&amp; A \\cos \\sqrt{- |K| } x + \\frac{B}{\\sqrt{- |K| } } \\sin \\sqrt{- |K| } x \\\\<br \/>\n&amp;=&amp; A \\cos(i \\sqrt{|K|} x)\u00a0 + \\frac{B}{i \\sqrt{|K|} } \\sin (i \\sqrt{|K|} x) \\\\<br \/>\n&amp;=&amp; A \\cosh(\\sqrt{|K|} x) + \\frac{B}{\\sqrt{|K|} } \\sinh (\\sqrt{|K|} x)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3060\u3057\uff0c$K=0$ \u306e\u5834\u5408\u306f<br \/>\n\\begin{eqnarray}<br \/>\ny &amp;=&amp; \\lim_{K\\rightarrow 0} \\left(A \\cos \\sqrt{K} x + \\frac{B}{\\sqrt{K} } \\sin \\sqrt{K} x \\right) \\\\<br \/>\n&amp;=&amp; A + B x<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3068\u306a\u308a\uff0c$K &gt; 0$ \u306e\u3068\u304d\u306e\u89e3\u3067 $K &lt; 0$ \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<h3>\u307e\u3068\u3081<\/h3>\n<p>\\( y^{&#8221;} +K y = 0\\) \u3042\u308b\u3044\u306f\u79fb\u9805\u3057\u3066 \\( y^{&#8221;} = -K y\\) \u306e\u89e3\u306f\uff0c\uff08\u7a4d\u5206\u5b9a\u6570\u3092 $A, B$ \u306b\u7f6e\u304d\u623b\u3057\u3066\uff09<\/p>\n<p>\\begin{eqnarray}<br \/>\nK &gt; 0 &amp;\\ \\ \\Longrightarrow \\ \\\u00a0 &amp; y = A \\cos(\\sqrt{K} \\ x) + B \\sin(\\sqrt{K} \\ x)\\\\<br \/>\nK &lt; 0 &amp;\\ \\ \\Longrightarrow \\ \\\u00a0 &amp; y = A \\cosh(\\sqrt{|K|} \\ x) + B \\sinh(\\sqrt{|K|} \\ x)\\\\<br \/>\nK = 0 &amp;\\ \\ \\Longrightarrow \\ \\\u00a0 &amp; y = A\u00a0 + B\\ x<br \/>\n\\end{eqnarray}<\/p>\n<h3>\u53c2\u8003\uff1a\u4e00\u822c\u7684\u306a\u6559\u79d1\u66f8\u306e\u8aac\u660e\u4f8b<\/h3>\n<p>\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<h4 id=\"yui_3_17_2_1_1645695245080_1514\">\\( K&gt;0 \\) \u306e\u5834\u5408<\/h4>\n<p>\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<p>$$ \\frac{d^2}{dx^2} e^{\\lambda x} = \\lambda^2 e^{\\lambda x} = -K e^{\\lambda x} $$<\/p>\n<p>\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<p>$$ y = C_1 e^{+i\\sqrt{K} x} + C_2 e^{-i\\sqrt{K} x} $$<\/p>\n<p>&#8230;\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<br \/>\n\\begin{eqnarray}<br \/>\ne^{+i \\sqrt{K} x} &amp;=&amp; \\cos(\\sqrt{K} x) + i \\sin(\\sqrt{K} x) \\\\<br \/>\ne^{-i \\sqrt{K} x} &amp;=&amp; \\cos(\\sqrt{K} x) &#8211; i \\sin(\\sqrt{K} x)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3067\u3042\u308b\u304b\u3089<br \/>\n\\begin{eqnarray}<br \/>\ny &amp;=&amp; C_1 e^{+i \\sqrt{K} x} + C_2 e^{-i\\sqrt{K} x} \\\\<br \/>\n&amp;=&amp; C_1 \\left( \\cos(\\sqrt{K} x) + i \\sin(\\sqrt{K} x)\\right) + C_2\\left( \\cos(\\sqrt{K} x) &#8211; i \\sin(\\sqrt{K} x)\\right)\\\\<br \/>\n&amp;=&amp; \\left( C_1 + C_2\\right) \\cos(\\sqrt{K} x) + i \\left( C_1 &#8211; C_2 \\right)\\sin(\\sqrt{K} x)<br \/>\n\\end{eqnarray}<\/p>\n<p>&nbsp;<\/p>\n<p>\u3053\u3053\u3067\u3042\u3089\u305f\u3081\u3066 \\( C_1 + C_2 \\Rightarrow C&#8217;_1, \\ \\ i \\left( C_1 &#8211; C_2 \\right) \\Rightarrow C&#8217;_2\\) \u3068\u7f6e\u304d\u76f4\u3059\u3068<br \/>\n$$y = C&#8217;_1 \\cos(\\sqrt{K} x) + C&#8217;_2 \\sin(\\sqrt{K} x)$$\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<h4>\\( K = 0 \\) \u306e\u5834\u5408<\/h4>\n<p>\\( y^{&#8221;} = 0 \\) \u3068\u306a\u308b\u306e\u3067\uff0c\u7c21\u5358\u306b\u89e3\u3051\u3066 \\( y = C_1\u00a0 + C_2\\, x\\)\u3002<\/p>\n<h4>\\( K &lt; 0 \\) \u306e\u5834\u5408<\/h4>\n<p>\u3053\u306e\u3068\u304d\u306f\uff0c\\( K = &#8211; |K|\\) \u3068\u306a\u308b\u304b\u3089\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f<\/p>\n<p>$$ y^{&#8221;} = |K| y$$<\/p>\n<p>\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<p>$$ \\frac{d^2}{dx^2} e^{\\lambda x} =\\lambda^2 e^{\\lambda x} =\u00a0 |K| e^{\\lambda x} $$<\/p>\n<p>\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<p>$$ y = C_1 e^{+ \\sqrt{|K|} x} + C_2 e^{-\\sqrt{|K|} x} $$<\/p>\n<p>&nbsp;<\/p>\n<h4>\u307e\u3068\u3081<\/h4>\n<p>\\( y^{&#8221;} +K y = 0\\) \u3042\u308b\u3044\u306f\u79fb\u9805\u3057\u3066 \\( y^{&#8221;} = -K y\\) \u306e\u89e3\u306f\uff0c\uff08\u7a4d\u5206\u5b9a\u6570\u3092\u3042\u3089\u305f\u3081\u3066 $A, \\, B$ \u3068\u3057\u3066\uff09<br \/>\n\\begin{eqnarray}<br \/>\nK &gt; 0 &amp;\\ \\ \\Longrightarrow \\ \\\u00a0 &amp; y = A \\cos(\\sqrt{K} \\ x) + B \\sin(\\sqrt{K} \\ x)\\\\<br \/>\nK &lt; 0 &amp;\\ \\ \\Longrightarrow \\ \\\u00a0 &amp; y = A e^{+\\sqrt{|K|} \\ x} + B e^{-\\sqrt{|K|}\\\u00a0 x}\\\\<br \/>\nK = 0 &amp;\\ \\ \\Longrightarrow \\ \\\u00a0 &amp; y = A\u00a0 + B x<br \/>\n\\end{eqnarray}<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\\( y^{&#8221;} +\u00a0 K y = 0\\) \u3042\u308b\u3044\u306f\u79fb\u9805\u3057\u3066 \\(y^{&#8221;} = &#8211; K y\\) \u3092\u89e3\u304f\u3002<\/p><p><a class=\"more-link btn\" 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