{"id":2276,"date":"2022-02-24T18:53:55","date_gmt":"2022-02-24T09:53:55","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2276"},"modified":"2025-05-20T13:43:30","modified_gmt":"2025-05-20T04:43:30","slug":"%e5%ae%9a%e6%95%b0%e4%bf%82%e6%95%b02%e9%9a%8e%e7%b7%9a%e5%bd%a2%e9%9d%9e%e5%90%8c%e6%ac%a1%e6%96%b9%e7%a8%8b%e5%bc%8f","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\/%e5%ae%9a%e6%95%b0%e4%bf%82%e6%95%b02%e9%9a%8e%e7%b7%9a%e5%bd%a2%e9%9d%9e%e5%90%8c%e6%ac%a1%e6%96%b9%e7%a8%8b%e5%bc%8f\/","title":{"rendered":"\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f"},"content":{"rendered":"<p>\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u3068\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u53f3\u8fba\u306b\u975e\u540c\u6b21\u9805 \\( R(x) \\) \u304c\u3042\u308b\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u3053\u3068\u3002<br \/>\n$$ y^{&#8221;} + 2 b y&#8217; + c y = R(x) $$<br \/>\n\u7279\u6b8a\u89e3 \\( y_s(x) \\) \u3092\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f $$ y^{&#8221;}(x) + 2 b y'(x) + c y(x) = 0 $$ \u306e1\u6b21\u72ec\u7acb\u306a\u57fa\u672c\u89e3 \\( y_1(x), \\ y_2(x) \\) \u304a\u3088\u3073\u305d\u308c\u3089\u304b\u3089\u3064\u304f\u3089\u308c\u308b\u30ed\u30f3\u30b9\u30ad\u30a2\u30f3 $$ W(x) \\equiv y_1(x) y&#8217;_2(x) -y&#8217;_1(x) y_2(x) $$ \u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b\u516c\u5f0f\u306f\uff0c<\/p>\n<p>$$ y_s(x) = y_2(x) \\int \\frac{R(x) y_1 (x)}{W(x)} dx -y_1(x) \\int \\frac{R(x) y_2(x)}{W(x)} dx$$<\/p>\n<p><!--more--><\/p>\n<hr \/>\n<h3>\u57fa\u672c\u89e3\uff0c\u7279\u6b8a\u89e3\uff0c\u4e00\u822c\u89e3<\/h3>\n<div><\/div>\n<div id=\"yui_3_17_2_1_1645696342392_1405\">\u3053\u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u4e00\u822c\u89e3<\/strong><\/span>\u306f<br \/>\n$$ y = C_1 y_1 + C_2 y_2 + y_s $$<br \/>\n\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/div>\n<div>\u3053\u3053\u3067\uff0c\\( y = C_1\u00a0 y_1 + C_2 y_2 \\) \u306e\u90e8\u5206\u306f\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f \\( y^{&#8221;} + 2 b y&#8217; + c y = 0 \\) \u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u57fa\u672c\u89e3<\/strong><\/span> \\(y_1, \\ y_2 \\) \u306e\u7dda\u5f62\u548c\u3067\u4e0e\u3048\u3089\u308c\u308b\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\u3067\u3042\u308a\uff0c\\( y_s \\) \u306f\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f \\( y^{&#8221;} + 2 b y&#8217; + c y = R(x)\\) \u306e<span style=\"font-family: helvetica, arial, sans-serif;\"><strong>\u7279\u6b8a\u89e3<\/strong><\/span>\u3002<\/div>\n<div><\/div>\n<div>\u7e70\u308a\u8fd4\u3059\u304c\uff0c\\(y_1, \\ y_2 \\) \u306f\u540c\u6b21\u65b9\u7a0b\u5f0f \\( y^{&#8221;} + 2 b y&#8217; + c y = 0 \\) \u306e\u57fa\u672c\u89e3\uff0c\u3064\u307e\u308a<\/div>\n<div>$$y^{&#8221;}_1 + 2 b y&#8217;_1 + c y_1 = 0, \\quad y^{&#8221;}_2 + 2 b y&#8217;_2 + c y_2 = 0$$<\/div>\n<div>\u4e00\u65b9\uff0c\\( y_s \\) \u306f\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f \\( y^{&#8221;} + 2 b y&#8217; + c y = R(x)\\) \u306e\u7279\u6b8a\u89e3\uff0c\u3064\u307e\u308a<\/div>\n<div>$$ y^{&#8221;}_s + 2 b y&#8217;_s + c y_s = R(x)$$<\/div>\n<div><\/div>\n<div>\u7d50\u5c40\uff0c\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u3053\u306e\u7279\u6b8a\u89e3 \\( y_s \\) \u3092\u3069\u306e\u3088\u3046\u306b\u3057\u3066\u6c42\u3081\u308b\u304b\uff0c\u3068\u3044\u3046\u3053\u3068\u306b\u5c3d\u304d\u308b\u3002<\/div>\n<div><\/div>\n<div>\u3053\u3053\u3067\u306f\uff0c\u30ed\u30f3\u30b9\u30ad\u30a2\u30f3\u3092\u4f7f\u3063\u3066\u7279\u6b8a\u89e3\u3092\u6c42\u3081\u308b\u516c\u5f0f\u3092\u7d39\u4ecb\u3057\u3066\u304a\u304f\u3002<\/div>\n<div><\/div>\n<h3>\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3\u3092\u6c42\u3081\u308b\u516c\u5f0f\uff08\u30ed\u30f3\u30b9\u30ad\u30a2\u30f3\u3092\u4f7f\u3063\u3066\uff09<\/h3>\n<div>\n<p>\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f $$ y^{&#8221;}(x) + 2 b y'(x) + c y(x) = R(x) $$ \u306e\u7279\u6b8a\u89e3 \\( y_s(x) \\) \u3092\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f $$ y^{&#8221;}(x) + 2 b y'(x) + c y(x) = 0 $$ \u306e1\u6b21\u72ec\u7acb\u306a\u57fa\u672c\u89e3 \\( y_1(x), \\ y_2(x) \\) \u304a\u3088\u3073\u305d\u308c\u3089\u304b\u3089\u3064\u304f\u3089\u308c\u308b\u30ed\u30f3\u30b9\u30ad\u30a2\u30f3 $$ W(x) \\equiv y_1(x) y&#8217;_2(x) -y&#8217;_1(x) y_2(x) $$ \u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b\u516c\u5f0f\u304c\u3042\u308b\u3002<\/p>\n<p>$$ y_s(x) = y_2(x) \\int \\frac{R(x) y_1 (x)}{W(x)} dx -y_1(x) \\int \\frac{R(x) y_2(x)}{W(x)} dx$$<\/p>\n<p>\u3053\u3053\u3067\u306f\uff0c\u3053\u306e\u5f0f\u304c\u78ba\u304b\u306b\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u3092\u76f4\u63a5\u4ee3\u5165\u3057\u3066\u78ba\u304b\u3081\u3066\u307f\u307e\u3059\u3002<\/p>\n<p>\u307e\u305a\uff0c\u524d\u63d0\u3068\u3057\u3066\uff0c\\( y_1(x), \\ y_2(x) \\) \u306f\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u306a\u306e\u3067<br \/>\n$$ y^{&#8221;}_1 + 2 b y&#8217;_1 + c y_1 = 0, \\quad y^{&#8221;}_2 + 2 b y&#8217;_2 + c y_2 = 0$$<\/p>\n<p>\\(y_s(x) \\) \u306e1\u968e\u5fae\u5206\u306f\uff0c<\/p>\n<p>\\begin{eqnarray}<br \/>\ny&#8217;_s(x) &amp;=&amp; y&#8217;_2(x) \\int \\frac{R(x) y_1 (x)}{W(x)} dx -y&#8217;_1(x) \\int \\frac{R(x) y_2(x)}{W(x)} dx \\\\<br \/>\n&amp;&amp; + y_2(x) \\ \\frac{R(x) y_1 (x)}{W(x)} -y_1(x) \\ \\frac{R(x) y_2(x)}{W(x)} \\\\<br \/>\n&amp;=&amp; y&#8217;_2(x) \\int \\frac{R(x) y_1 (x)}{W(x)} dx -y&#8217;_1(x) \\int \\frac{R(x) y_2(x)}{W(x)} dx<br \/>\n\\end{eqnarray}<\/p>\n<p>\\(y_s(x) \\) \u306e2\u968e\u5fae\u5206\u306f\uff0c<\/p>\n<p>\\begin{eqnarray} y^{&#8221;}_s(x) &amp;=&amp; y^{&#8221;}_2(x) \\int \\frac{R(x) y_1 (x)}{W(x)} dx -y^{&#8221;}_1(x) \\int \\frac{R(x) y_2(x)}{W(x)} dx \\\\<br \/>\n&amp;&amp; + y&#8217;_2(x) \\ \\frac{R(x) y_1 (x)}{W(x)} -y&#8217;_1(x) \\ \\frac{R(x) y_2(x)}{W(x)} \\\\<br \/>\n&amp;=&amp; y^{&#8221;}_2(x) \\int \\frac{R(x) y_1 (x)}{W(x)} dx -y^{&#8221;}_1(x) \\int \\frac{R(x) y_2(x)}{W(x)} dx + R(x)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u307e\u3068\u3081\u3066\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u5de6\u8fba\u306b\u4ee3\u5165\u3059\u308b\u3068\uff0c<\/p>\n<p>\\begin{eqnarray} y^{&#8221;}_s + 2 b y&#8217;_s + c y_s &amp;= &amp;<br \/>\n\\left(y^{&#8221;}_2 + 2 b y&#8217;_2 + c y_2 \\right) \\int \\frac{R(x) y_1 (x)}{W(x)} dx \\\\<br \/>\n&amp;&amp; -\\left(y^{&#8221;}_1 + 2 b y&#8217;_1 + c y_1 \\right) \\int \\frac{R(x) y_2(x)}{W(x)} dx + R(x) \\\\ &amp;=&amp; R(x)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u3068\u306a\u3063\u3066\uff0c\u78ba\u304b\u306b\u7279\u6b8a\u89e3\u306b\u306a\u3063\u3066\u3044\u308b\uff01<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u3068\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u53f3\u8fba\u306b\u975e\u540c\u6b21\u9805 \\( R(x) \\) \u304c\u3042\u308b\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u3053\u3068\u3002 $$ y^{&#8221;} + 2 b y&#8217; + c y = R(x) $$ \u7279\u6b8a\u89e3 \\( y_s(x) \\) \u3092\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f $$ y^{&#8221;}(x) + 2 b y'(x) + c y(x) = 0 $$ \u306e1\u6b21\u72ec\u7acb\u306a\u57fa\u672c\u89e3 \\( y_1(x), \\ y_2(x) \\) \u304a\u3088\u3073\u305d\u308c\u3089\u304b\u3089\u3064\u304f\u3089\u308c\u308b\u30ed\u30f3\u30b9\u30ad\u30a2\u30f3 $$ W(x) \\equiv y_1(x) y&#8217;_2(x) -y&#8217;_1(x) y_2(x) $$ \u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b\u516c\u5f0f\u306f\uff0c<\/p>\n<p>$$ y_s(x) = y_2(x) \\int \\frac{R(x) y_1 (x)}{W(x)} dx -y_1(x) \\int \\frac{R(x) y_2(x)}{W(x)} dx$$<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":2224,"menu_order":11,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2276","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.0.1 - aioseo.com -->\n\t<meta name=\"description\" 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