{"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":"

\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
\n$$ y^{”} + 2 b y’ + c y = R(x) $$
\n\u7279\u6b8a\u89e3 \\( y_s(x) \\) \u3092\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f $$ y^{”}(x) + 2 b y'(x) + c y(x) = 0 $$ \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 $$ W(x) \\equiv y_1(x) y’_2(x) -y’_1(x) y_2(x) $$ \u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b\u516c\u5f0f\u306f\uff0c<\/p>\n

$$ y_s(x) = y_2(x) \\int \\frac{R(x) y_1 (x)}{W(x)} dx -y_1(x) \\int \\frac{R(x) y_2(x)}{W(x)} dx$$<\/p>\n

<\/p>\n

\n

\u57fa\u672c\u89e3\uff0c\u7279\u6b8a\u89e3\uff0c\u4e00\u822c\u89e3<\/h3>\n
<\/div>\n
\u3053\u306e\u4e00\u822c\u89e3<\/strong><\/span>\u306f
\n$$ y = C_1 y_1 + C_2 y_2 + y_s $$
\n\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/div>\n
\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^{”} + 2 b y’ + c y = 0 \\) \u306e\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^{”} + 2 b y’ + c y = R(x)\\) \u306e\u7279\u6b8a\u89e3<\/strong><\/span>\u3002<\/div>\n
<\/div>\n
\u7e70\u308a\u8fd4\u3059\u304c\uff0c\\(y_1, \\ y_2 \\) \u306f\u540c\u6b21\u65b9\u7a0b\u5f0f \\( y^{”} + 2 b y’ + c y = 0 \\) \u306e\u57fa\u672c\u89e3\uff0c\u3064\u307e\u308a<\/div>\n
$$y^{”}_1 + 2 b y’_1 + c y_1 = 0, \\quad y^{”}_2 + 2 b y’_2 + c y_2 = 0$$<\/div>\n
\u4e00\u65b9\uff0c\\( y_s \\) \u306f\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f \\( y^{”} + 2 b y’ + c y = R(x)\\) \u306e\u7279\u6b8a\u89e3\uff0c\u3064\u307e\u308a<\/div>\n
$$ y^{”}_s + 2 b y’_s + c y_s = R(x)$$<\/div>\n
<\/div>\n
\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>\n
\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>\n

\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
\n

\u5b9a\u6570\u4fc2\u65702\u968e\u7dda\u5f62\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f $$ y^{”}(x) + 2 b y'(x) + c y(x) = R(x) $$ \u306e\u7279\u6b8a\u89e3 \\( y_s(x) \\) \u3092\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f $$ y^{”}(x) + 2 b y'(x) + c y(x) = 0 $$ \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 $$ W(x) \\equiv y_1(x) y’_2(x) -y’_1(x) y_2(x) $$ \u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b\u516c\u5f0f\u304c\u3042\u308b\u3002<\/p>\n

$$ y_s(x) = y_2(x) \\int \\frac{R(x) y_1 (x)}{W(x)} dx -y_1(x) \\int \\frac{R(x) y_2(x)}{W(x)} dx$$<\/p>\n

\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

\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
\n$$ y^{”}_1 + 2 b y’_1 + c y_1 = 0, \\quad y^{”}_2 + 2 b y’_2 + c y_2 = 0$$<\/p>\n

\\(y_s(x) \\) \u306e1\u968e\u5fae\u5206\u306f\uff0c<\/p>\n

\\begin{eqnarray}
\ny’_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 \\\\
\n&& + y_2(x) \\ \\frac{R(x) y_1 (x)}{W(x)} -y_1(x) \\ \\frac{R(x) y_2(x)}{W(x)} \\\\
\n&=& 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
\n\\end{eqnarray}<\/p>\n

\\(y_s(x) \\) \u306e2\u968e\u5fae\u5206\u306f\uff0c<\/p>\n

\\begin{eqnarray} 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 \\\\
\n&& + y’_2(x) \\ \\frac{R(x) y_1 (x)}{W(x)} -y’_1(x) \\ \\frac{R(x) y_2(x)}{W(x)} \\\\
\n&=& 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 + R(x)
\n\\end{eqnarray}<\/p>\n

\u307e\u3068\u3081\u3066\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u5de6\u8fba\u306b\u4ee3\u5165\u3059\u308b\u3068\uff0c<\/p>\n

\\begin{eqnarray} y^{”}_s + 2 b y’_s + c y_s &= &
\n\\left(y^{”}_2 + 2 b y’_2 + c y_2 \\right) \\int \\frac{R(x) y_1 (x)}{W(x)} dx \\\\
\n&& -\\left(y^{”}_1 + 2 b y’_1 + c y_1 \\right) \\int \\frac{R(x) y_2(x)}{W(x)} dx + R(x) \\\\ &=& R(x)
\n\\end{eqnarray}<\/p>\n

\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":"

\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 $$ y^{”} + 2 b y’ + c y = R(x) $$ \u7279\u6b8a\u89e3 \\( y_s(x) \\) \u3092\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f $$ y^{”}(x) + 2 b y'(x) + c y(x) = 0 $$ \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 $$ W(x) \\equiv y_1(x) y’_2(x) -y’_1(x) y_2(x) $$ \u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b\u516c\u5f0f\u306f\uff0c<\/p>\n

$$ y_s(x) = y_2(x) \\int \\frac{R(x) y_1 (x)}{W(x)} dx -y_1(x) \\int \\frac{R(x) y_2(x)}{W(x)} dx$$<\/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":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2276","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=2276"}],"version-history":[{"count":16,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2276\/revisions"}],"predecessor-version":[{"id":10235,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2276\/revisions\/10235"}],"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=2276"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}