{"id":6312,"date":"2023-05-16T09:53:15","date_gmt":"2023-05-16T00:53:15","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=6312"},"modified":"2025-05-20T13:42:37","modified_gmt":"2025-05-20T04:42:37","slug":"%e9%9d%9e%e5%90%8c%e6%ac%a1%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%ae%e7%89%b9%e6%ae%8a%e8%a7%a3%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b%e4%be%8b%e9%a1%8c","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\/%e9%9d%9e%e5%90%8c%e6%ac%a1%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%ae%e7%89%b9%e6%ae%8a%e8%a7%a3%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b%e4%be%8b%e9%a1%8c\/","title":{"rendered":"\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3\u3092\u6c42\u3081\u308b\u4f8b\u984c"},"content":{"rendered":"

<\/p>\n

\u4f8b\u984c 1<\/h3>\n

\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f \\( y^{\\prime\\prime} + y^{\\prime} -2 y = 2 e^{-x} \\) \u306e\u7279\u6b8a\u89e3\u3092\u6c42\u3081\u308b\u3002<\/p>\n

\u307e\u305a\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u57fa\u672c\u89e3<\/h4>\n

\u540c\u6b21\u65b9\u7a0b\u5f0f \\(y^{\\prime\\prime} + y^{\\prime} -2 y=0\\) \u3092 \\(y^{\\prime\\prime}+2 b \u00a0 y^{\\prime} + c y =0 \\) \u306e\u5f62\u306b\u3042\u3066\u306f\u3081\u3066<\/p>\n

$$ b = \\frac{1}{2}, \\ c = -2, \\ c -b^2 = -2 -\\frac{1}{4} = -\\frac{9}{4} < 0$$<\/p>\n

\u3067\u3042\u308b\u304b\u3089\uff0c\u3053\u306e\u3042\u305f\u308a<\/strong><\/span><\/a>\u3092\u53c2\u7167\u3057\u3066<\/p>\n

\\begin{eqnarray}
\ny &=& e^{-b x} \\left\\{A \\cosh\\sqrt{b^2-c}\\,x + B \\sinh\\sqrt{b^2-c}\\,x \\right\\} \\\\
\n&=& e^{-\\frac{1}{2} x} \\left\\{A \\cosh\\frac{3}{2} x + B \\sinh\\frac{3}{2}x \\right\\}
\n\\end{eqnarray}<\/p>\n

\u3068\u3057\u3066\u3082\u3088\u3044\u304c\uff0c\u3055\u3089\u306b<\/p>\n

\\begin{eqnarray}
\ny &=& e^{-\\frac{1}{2} x} \\left\\{A \\cosh\\frac{3}{2} x + B \\sinh\\frac{3}{2}x \\right\\} \\\\
\n&=&e^{-\\frac{1}{2} x} \\left\\{A \\frac{e^{\\frac{3}{2}x} + e^{-\\frac{3}{2}x}}{2} + B \\frac{e^{\\frac{3}{2}x} -e^{-\\frac{3}{2}x}}{2} \\right\\} \\\\
\n&=& \\frac{A}{2} \\left(e^x + e^{-2 x} \\right)+ \\frac{B}{2} \\left(e^x -e^{-2 x} \\right)\\\\
\n&=& \\frac{A-B}{2} e^{-2x} + \\frac{A+B}{2} e^x \\\\
\n&=& C_1 e^{-2x} + C_2 e^x
\n\\end{eqnarray}<\/p>\n

\u3064\u307e\u308a\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u57fa\u672c\u89e3\u3092<\/p>\n

\\begin{eqnarray}
\ny_1 &=& e^{-2 x} \\\\
\ny_2 &=& e^{ x}
\n\\end{eqnarray}<\/p>\n

\u3068\u3059\u308b\u307b\u3046\u304c\u7c21\u5358\u306b\u8a08\u7b97\u304c\u9032\u3080\u3002<\/p>\n

\u3053\u306e\u57fa\u672c\u89e3\u3092\u6c42\u3081\u308b\u5225\u306e\u65b9\u6cd5\u3068\u3057\u3066\uff0c\\( y = e^{\\lambda x} \\) \u3068\u304a\u304f\u3068\uff0c\\(\\lambda\\) \u306f\u4ee5\u4e0b\u3092\u6e80\u305f\u3059\u3002
\n$$ \\lambda^2 + \\lambda -2 = 0 $$ \u56e0\u6570\u5206\u89e3\u3067\u304d\u3066\uff0c
\n$$ (\\lambda + 2)(\\lambda -1) = 0, \\quad \\therefore \\lambda = -2, 1 $$<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u57fa\u672c\u89e3\u306f<\/p>\n

\\begin{eqnarray}
\ny_1 &=& e^{-2 x} \\\\
\ny_2 &=& e^{ x}
\n\\end{eqnarray}<\/p>\n

\u3068\u3057\u3066\u3082\u3088\u3044\u3060\u308d\u3046\u3002<\/p>\n

\u30ed\u30f3\u30b9\u30ad\u30a2\u30f3<\/h4>\n

\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u57fa\u672c\u89e3\u3092\u4f7f\u3046\u3068\uff0c\u30ed\u30f3\u30b9\u30ad\u30a2\u30f3 \\(W(x) \\) \u306f<\/p>\n

\\begin{eqnarray}
\nW(x) &=& y_1(x) \\, y’_2(x) -y’_1(x)\\, y_2(x) \\\\
\n&=& e^{-2 x} e^{ x}-(-2 e^{-2 x}) e^{ x}\\\\
\n&=& 3 e^{-x}
\n\\end{eqnarray}<\/p>\n

\u3042\u3068\u306f\u516c\u5f0f\u306b\u5165\u308c\u3066…<\/h4>\n

\u5f93\u3063\u3066\uff0c\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3\u306f<\/p>\n

\\begin{eqnarray}
\ny_s &=& y_2(x)\\int \\frac{R(x) y_1(x)}{W(x)} dx -y_1(x) \\int\u00a0 \\frac{R(x) y_2(x)}{W(x)} dx\\\\
\n&=& e^x \\int \\frac{2 e^{-x} \\,e^{-2x}}{3 e^{-x} }dx-
\ne^{-2 x} \\int\\frac{2 e^{-x} \\,e^{ x}}{3 e^{-x}}dx \\\\
\n&=& \\frac{2}{3} e^x \\int e^{-2 x} \\,dx -\\frac{2}{3} e^{-2x} \\int e^x \\,dx \\\\
\n&=& -e^{-x}
\n\\end{eqnarray}<\/p>\n

\u6700\u5f8c\u306b\uff0c\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\u306f…<\/h4>\n

\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u57fa\u672c\u89e3\u306e\u7dda\u5f62\u548c\u3067\u3042\u308b\u4e00\u822c\u89e3\u3068\uff0c\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3\u306e\u548c\u3060\u304b\u3089
\n$$ y = C_1 y_1 + C_2 y_2 + y_s = C_1 e^{-2x} + C_2 e^x -e^{-x} $$<\/p>\n

\u4f8b\u984c 2<\/h3>\n

\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f \\( y^{\\prime\\prime} + a^2 y = \\sin a x\\) \u306e\u7279\u6b8a\u89e3\u3092\u6c42\u3081\u308b\u3002<\/p>\n

\u307e\u305a\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u57fa\u672c\u89e3<\/h4>\n

\u540c\u6b21\u65b9\u7a0b\u5f0f \\( y^{\\prime\\prime} + a^2 y = 0 \\) \u306e1\u6b21\u72ec\u7acb\u306a\u57fa\u672c\u89e3\u306f\u810a\u9ac4\u53cd\u5c04\u3067<\/strong><\/span><\/a>\uff0c
\n$$ y_1 = \\cos a x, \\quad y_2 = \\sin a x $$<\/p>\n

\u30ed\u30f3\u30b9\u30ad\u30a2\u30f3<\/h4>\n

$$ W = y_1 y’_2 -y’_1 y_2 = \\cos a x \\cdot a \\cos ax -(-a) \\sin a x \\cdot \\sin a x = a $$<\/p>\n

\u3042\u3068\u306f\u516c\u5f0f\u306b\u5165\u308c\u3066…<\/h4>\n

\u975e\u540c\u6b21\u9805\u306f \\( R(x) = \\sin a x \\) \u3060\u304b\u3089
\n<\/span><\/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&=& \\sin a x \\int \\frac{\\sin a x\\ \\cos a x}{a} dx -\\cos a x \\int \\frac{\\sin a x \\ \\sin a x}{a} dx \\\\
\n&=& \\frac{1}{2a} \\sin a x \\int \\sin 2 a x \\, dx -\\frac{1}{2a} \\cos a x \\int (1 -\\cos 2 a x) \\, dx \\\\
\n&=& -\\frac{1}{4a^2} \\sin a x \\ \\cos 2 a x -\\frac{1}{2 a} \\cos a x \\left( x -\\frac{1}{2a} \\sin 2 a x\\right) \\\\
\n&=& -\\frac{x}{2 a} \\cos a x + \\frac{1}{4 a^2}\\left( \\sin 2 a x \\ \\cos a x -\\cos 2 a x \\ \\sin a x \\right) \\\\
\n&=& -\\frac{x}{2 a} \\cos a x + \\frac{1}{4 a^2} \\sin a x \\end{eqnarray}<\/p>\n

\u6700\u5f8c\u306e \\( \\sin a x \\) \u306b\u6bd4\u4f8b\u3059\u308b\u9805\u306f\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u57fa\u672c\u89e3 \\( y_2 = \\sin a x \\) \u306b\u6bd4\u4f8b\u3059\u308b\u9805\u305d\u306e\u3082\u306e\u3067\u3042\u308b\u304b\u3089\uff0c\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\u306b\u7d44\u307f\u8fbc\u307e\u308c\u308b\u9805\u3067\u3042\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3\u3068\u3057\u3066\u306f
\n$$ y_s = -\\frac{x}{2 a} \\cos a x $$<\/p>\n

$y_s$ \u306b\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u57fa\u672c\u89e3\u306b\u6bd4\u4f8b\u3059\u308b\u9805\u304c\u307e\u304e\u308c\u3053\u3080\u4e0d\u5b9a\u6027\u306f\uff0c\u4e0d\u5b9a\u7a4d\u5206\u306e\u4e0d\u5b9a\u6027\u306b\u8d77\u56e0\u3059\u308b\u3002\u305f\u3081\u3057\u306b\uff0c<\/p>\n

\\begin{eqnarray}
\n\\int \\frac{\\sin a x\\ \\cos a x}{a} dx &=&\\frac{1}{2a}\u00a0 \\int \\sin 2 a x \\, dx \\\\
\n&=& -\\frac{1}{4 a^2} \\cos 2 a x
\n\\end{eqnarray}<\/p>\n

\u306e\u304b\u308f\u308a\u306b<\/p>\n

\\begin{eqnarray}
\n{\\color{blue}{\\int \\frac{\\sin a x\\ \\cos a x}{a} dx}}
\n&=& {\\color{blue}{-\\frac{1}{2 a^2} \\cos^2 a x}}
\n\\end{eqnarray}<\/p>\n

\u3092\u4f7f\u3046\u3068\uff0c\uff08\u4e0a\u5f0f\u306e\u4e21\u8fba\u3092 $x$ \u3067\u5fae\u5206\u3059\u308c\u3070\u6b63\u3057\u3044\u3053\u3068\u304c\u308f\u304b\u308b\u3088\u306d\uff09<\/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&=& \\sin a x {\\color{blue}{\\int \\frac{\\sin a x\\ \\cos a x}{a} dx}} -\\cos a x \\int \\frac{\\sin a x \\ \\sin a x}{a} dx \\\\
\n&=&{\\color{blue}{ -\\frac{1}{2 a^2} }}\\sin a x \\ {\\color{blue}{\\cos^2 a x}} -\\frac{1}{2 a} \\cos a x \\left( x -\\frac{1}{2a} \\sin 2 a x\\right) \\\\
\n&=& -\\frac{x}{2 a} \\cos a x + \\frac{1}{4 a^2}\\left( \\sin 2 a x \\ \\cos a x -{\\color{blue}{2\\cos^2 a x}} \\ \\sin a x \\right) \\\\
\n&=& -\\frac{x}{2 a} \\cos a x + \\frac{1}{4 a^2}\\left( 2 \\sin\u00a0 a x\u00a0 \\cos a x\\ \\cos a x -{\\color{blue}{2\\cos^2 a x}} \\ \\sin a x \\right) \\\\&=& -\\frac{x}{2 a} \\cos a x \\end{eqnarray}<\/p>\n

\u3068\u306a\u308a\uff0c\u3081\u3067\u305f\u304f\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3\u306e\u307f\u304c\u3067\u3066\u304f\u308b\u3002<\/p>\n

\u6700\u5f8c\u306b\uff0c\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\u306f…<\/h4>\n

\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u57fa\u672c\u89e3\u306e\u7dda\u5f62\u548c\u3067\u3042\u308b\u4e00\u822c\u89e3\u3068\uff0c\u975e\u540c\u6b21\u65b9\u7a0b\u5f0f\u306e\u7279\u6b8a\u89e3\u306e\u548c\u3060\u304b\u3089
\n$$ y = C_1 y_1 + C_2 y_2 + y_s = C_1 \\cos a x + C_2 \\sin a x -\\frac{x}{2 a} \\cos a x $$<\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2276,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-6312","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\/6312","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\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=6312"}],"version-history":[{"count":12,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6312\/revisions"}],"predecessor-version":[{"id":10234,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6312\/revisions\/10234"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2276"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=6312"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}