{"id":129,"date":"2025-06-24T23:53:07","date_gmt":"2025-06-24T15:53:07","guid":{"rendered":"http:\/\/116.62.30.57\/?p=129"},"modified":"2025-09-14T23:19:19","modified_gmt":"2025-09-14T15:19:19","slug":"%e5%be%ae%e5%88%86%e7%ae%97%e5%ad%90%e6%b3%95","status":"publish","type":"post","link":"https:\/\/qiuqiqaq.fun\/index.php\/2025\/06\/24\/%e5%be%ae%e5%88%86%e7%ae%97%e5%ad%90%e6%b3%95\/","title":{"rendered":"\u5fae\u5206\u7b97\u5b50\u6cd5"},"content":{"rendered":"\n

\u6211\u4eec\u5728\u5c0f\u5b66\u4e00\u5e74\u7ea7\u5c31\u5b66\u8fc7\uff0c\u4e8c\u9636\u5e38\u7cfb\u6570\u975e\u9f50\u6b21\u7ebf\u6027\u5fae\u5206\u65b9\u7a0b\uff1a<\/p>\n\n\n\n

$$y”+py’+q=f(x)$$<\/p>\n\n\n\n

\u89e3\u7684\u7ed3\u6784\u4e3a\u9f50\u6b21\u7ebf\u6027\u5fae\u5206\u65b9\u7a0b\u7684\u89e3$\\sideset{}{_\u9f50}y$\u52a0\u4e0a\u7279\u89e3y*,\u5373\uff1a<\/p>\n\n\n\n

$$y = \\sideset{}{_\u9f50}y+y*$$<\/p>\n\n\n\n

\u5bf9\u4e8e\u9f50\u6b21\u7ebf\u6027\u5fae\u5206\u65b9\u7a0b\u7684\u89e3\u6211\u4eec\u6682\u4e14\u6309\u4e0b\u4e0d\u8868\uff0c\u4eca\u5929\u8bb2\u7279\u89e3y*\u7684\u6c42\u6cd5\u3002\u7279\u89e3\u5e38\u7528\u7684\u89e3\u6cd5\u6709\u4e24\u79cd\uff0c\u5206\u522b\u4e3a\uff1a<\/p>\n\n\n\n

\n

1\u3001\u5f85\u5b9a\u7cfb\u6570\u6cd5<\/p>\n<\/blockquote>\n\n\n\n

\n

2\u3001\u5fae\u5206\u7b97\u5b50\u6cd5<\/p>\n<\/blockquote>\n\n\n\n

\u5f85\u5b9a\u7cfb\u6570\u6cd5<\/strong>\u867d\u7136\u7b80\u5355\u5bb9\u6613\u7406\u89e3\uff0c\u4f46\u662f\u8ba1\u7b97\u91cf\u5927\uff0c\u6548\u7387\u4e0d\u6562\u606d\u7ef4\u3002\u5fae\u5206\u7b97\u5b50\u6cd5<\/strong>\u5bf9\u4e8e\u521a\u63a5\u89e6\u7684\u4eba\u6765\u8bf4\u53ef\u80fd\u4e0d\u592a\u5bb9\u6613\u7406\u89e3\uff0c\u672c\u4eba\u5f53\u521d\u7b2c\u4e00\u6b21\u542c\u5f20\u5b87\u8bb2\u7684\u65f6\u5019\u542c\u7761\u7740\u4e86\ud83d\ude05\u3002\u7531\u4e8e\u6700\u8fd1\u505a\u9898\u7528\u5f85\u5b9a\u7cfb\u6570\u6cd5<\/strong>\u7b97\u70e6\u4e86\uff0c\u8fd8\u5bb9\u6613\u9519\uff0c\u4e8e\u662f\u4e4e\u6076\u8865\u4e86\u4e00\u4e0b\u5fae\u5206\u7b97\u5b50\u6cd5<\/strong>\u5e76\u7a0d\u5fae\u603b\u7ed3\u4e86\u4e00\u4e0b\u3002<\/p>\n\n\n\n

\u5b9a\u4e49<\/h2>\n\n\n\n

\u9996\u5148\uff0c\u6211\u4eec\u5b9a\u4e49\u7b26\u53f7”D”\u8868\u793a\u6c42\u5bfc\uff0c\u4f8b\u5982\uff1a<\/p>\n\n\n\n

$$ Dsinx = cosx \\quad\\quad Dx = 1 \\quad\\quad (D+1)x^{2} = 2x + x^{2} $$<\/p>\n\n\n\n

“D”\u8868\u793a\u6c42\u5bfc\uff0c\u90a3\u4e48\u5f88\u81ea\u7136\u7684\uff0c\u5b9a\u4e49”$\\frac{1}{D}$”\u8868\u793a\u79ef\u5206\uff0c\u4f8b\u5982\uff1a<\/p>\n\n\n\n

$$\\frac{1}{D}sinx = -cosx \\quad\\quad \\frac{1}{D}x = \\frac{1}{2} x^{2} $$<\/p>\n\n\n\n

\u6c42\u89e3\u6b65\u9aa4<\/h2>\n\n\n\n

\u5bf9\u4e8e\u5fae\u5206\u65b9\u7a0b\uff1a$$y”+ay’+b=f(x)$$<\/p>\n\n\n\n

\u6211\u4eec\u5148\u5199\u51fa\u5176\u7279\u5f81\u65b9\u7a0b\uff1a$\\lambda ^{2} +a\\lambda+b = 0$,\u7136\u540e\u8bb0\uff1a$$F(D)= D^{2}+aD+b$$<\/p>\n\n\n\n

\u6b64\u65f6\uff0c\u7279\u89e3\uff1a$$y*=\\frac{1}{F(D)} f(x)=\\frac{1}{D^{2}+aD+b} f(x)$$\u5bf9\u5e94\u4e0d\u540c\u7c7b\u578b\u7684\u65b9\u7a0b\uff0c\u7528\u76f8\u5e94\u7684\u65b9\u6cd5\u6d88\u53bbD\u5373\u53ef\u6c42\u7684\u89e3\u3002\u4e3b\u8981\u6709\u4ee5\u4e0b\u56db\u79cd\u7c7b\u578b\u3002<\/p>\n\n\n\n

\u7c7b\u578b\u4e00:$f(x)=e^{kx}$<\/h2>\n\n\n\n

\u8be5\u7c7b\u578b\u5c06\u7cfb\u6570K\u5e26\u5165F(D)\u4e2d\u7684D\u8ba1\u7b97\u5373\u53ef\uff0c\u82e5\u5e26\u5165\u540eF(D)=0,\u5bf9F(D)\u8fdb\u884c\u6c42\u5bfc\u5e76\u63d0\u51fa\u4e00\u4e2ax\uff0c\u7136\u540e\u63a5\u7740\u5e26\u5165\uff0c\u82e5\u4ee3\u5165K\u540eF'(D)\u8fd8\u4e3a0,\u518d\u6b21\u8fdb\u884c\u6c42\u5bfc\u5e76\u63d0\u51fa\u4e00\u4e2ax\u5373\u53ef\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u4e0b\u9762\u7684\u4f8b\u9898\u6765\u7406\u89e3\uff1a<\/p>\n\n\n\n

\"\"<\/div><\/figure>\n\n\n\n

\u603b\u7ed3\uff1a\u5bf9\u4e8e\u8be5\u7c7b\u578b\uff0c\u5c06K\u4ee3\u5165D\u8ba1\u7b97\uff0c\u82e5F(D)\u4e3a\u96f6\uff0c\u5bf9F(D)\u6c42\u5bfc\u5e76\u63d0\u51fa\u4e00\u4e2aX,\u82e5\u6c42\u5bfc\u4e00\u6b21\u540e\u4ee3\u5165K\u8ba1\u7b97\u7ed3\u679c\u4ecd\u7136\u4e3a0\uff0c\u5219\u518d\u6c42\u5bfc\u63d0X\uff0c\u4e00\u822c\u8003\u7814\u6570\u5b66\u9898\u76ee\u5230\u6b64\u5373\u53ef\u7ed3\u675f\u4e86\u3002<\/p>\n\n\n\n

\u7c7b\u578b\u4e8c:$f(x)=sinax\/cosax$<\/h2>\n\n\n\n

\u6b64\u7c7b\u578b\u7684\u89e3\u6cd5\u4e3a\u5c06a\u4e58\u4ee5\u865a\u6570i\uff0c\u5373ai<\/strong>(\u6ce8\uff1a$(ai)^{2}=-a^{2}$)\u5e26\u5165D,\u6839\u636e\u7279\u5f81\u65b9\u7a0b\u4e0d\u540c\u53c8\u53ef\u4ee5\u5206\u4e3a\u4e24\u7c7b\u3002<\/p>\n\n\n\n

1\u3001\u5f62\u5982$F(D)=D^{2} +q$
\u82e5$ F(D)=D^{2} +q\\sideset{}{_{D=ai}}|\\ne 0 $\uff0c\u6b64\u65f6\uff0c$$y^{*}=\\frac{1}{D^{2}+q\\sideset{}{_{D=ai}}|} f(x)$$<\/p>\n\n\n\n

2\u3001\u82e5$ F(D)=D^{2} +q\\sideset{}{_{D=ai}}|= 0 $\uff0c\u9700\u8981\u5bf9F(D)\u6c42\u5bfc\uff0c\u5e76\u63d0\u51fax,\u518d\u5bf9f(x)\u8fdb\u884c\u79ef\u5206\uff0c\u6b64\u65f6
$$y^{*}=x\\frac{1}{(D^{2}+q)’} f(x)$$<\/p>\n\n\n\n

\u6211\u4eec\u901a\u8fc7\u4e24\u4e2a\u4f8b\u9898\u7a0d\u52a0\u7406\u89e3\uff1a<\/p>\n\n\n\n

\u4f8b\u9898\u4e00\uff1a$y”-y’=sinx$, \u6c42$y*$ <\/p>\n\n\n\n

\u89e3: $F(D)=D^{2}-1$\uff0c\u5c06$D=(i)$\u4ee3\u5165\u5f97\uff1a <\/p>\n\n\n\n

$$F(D)=D^{2} +q\\sideset{}{_{D=i}}|=-1-1=-2\\ne 0$$\u4e8e\u662f\u4e4e\uff0c<\/em><\/p>\n\n\n\n

$$y^{*}=\\frac{1}{D^{2}-1\\sideset{}{_{D=i}}|} f(x)=-\\frac{1}{2}sinx$$<\/p>\n\n\n\n

\u4f8b\u9898\u4e8c\uff1a$y”+4y’=cos2x$, \u6c42$y*$ <\/p>\n\n\n\n

\u89e3\uff1a\u89c2\u5bdf\u5230$$F(D)=D^{2} +q\\sideset{}{_{D=2i}}|=-4+4=0$$\uff0c\u4e8e\u662f\uff0c\u6211\u4eec\u5bf9F(D)\u6c42\u5bfc\u5e76\u63d0\u51fa\u4e00\u4e2ax,\u5f97\u5230$x\\frac{1}{2D} cos2x$\uff0c\u518d\u5bf9$f(x)$\u79ef\u5206\u5373\u53ef($\\frac{1}{D}$\u8868\u793a\u79ef\u5206\uff09\uff0c\u53ef\u4ee5\u89e3\u5f97\uff1a$$y^{*}=x\\frac{1}{F(D)’} f(x)=x\\frac{1}{2D} cos2x=\\frac{x}{4} sin2x$$<\/p>\n\n\n\n

1\u3001\u5f62\u5982$F(D)=D^{2} +pD+q$<\/p>\n\n\n\n

\u5bf9\u4e8e\u6b64\u7c7b\u578b\uff0c\u7528$ai$\u5e26\u5165$D^{2}$\u540e\uff0c\u4f1a\u5f97\u5230$\\frac{1}{pD+\\alpha }$($\\alpha$\u4e3a\u5e38\u6570)\u7684\u7ec4\u5408\uff0c\u5bf9\u4e8e\u8fd9\u4e2a\u7ec4\u5408\u6211\u4eec\u662f\u65e0\u6cd5\u5904\u7406\u7684\uff0c\u6240\u4ee5\u8981\u60f3\u529e\u6cd5\u5c06\u5176\u6d88\u53bb\uff0c\u8fd9\u4e2a\u65f6\u5019\u5c31\u9700\u8981\u7528\u5230\u5e73\u65b9\u5dee\u516c\u5f0f\uff0c\u5c06\u5206\u6bcd\u7684$\\frac{1}{pD+\\alpha }$\u6d88\u53bb\u51d1\u51fa$D^{2}$\uff0c\u7136\u540e\u7ee7\u7eed\u5e26\u5165$ai$\u3002\u5982\u4e0b\u9762\u4f8b\u9898\u3002<\/p>\n\n\n\n

\u4f8b\u4e09\uff1a$y”+3y’-2y=sin2x$ \u6c42$y*$<\/p>\n\n\n\n

$y*=\\frac{1}{D^{2}+3D-2}sin2x$\u5c062i\u4ee3\u5165\u5f97$\\Longrightarrow \\frac{1}{3D-6}sin2x$,\u6b64\u65f6\uff0c\u7528\u5e73\u65b9\u5dee\u516c\u5f0f\u5904\u7406$ \\frac{1}{3D-6}$,$$y*=\\frac{1}{3D-6}sin2x = \\frac{1}{3}\\frac{D+2}{D^{2}-4}sin2x=-\\frac{1}{24}(D+2)sin2x=-\\frac{1}{12}(sin2x+cos2x)$$<\/p>\n\n\n\n

\u7c7b\u578b\u4e09:$f(x)=e^{kx} \\cdot \\varphi (x)$($\\varphi (x)$\u4e3a$sinax,cosax,x\u7684\u591a\u9879\u5f0f$)<\/h2>\n\n\n\n

\u6b64\u7c7b\u578b\u6c42\u89e3\u9700\u8981\u6211\u4eec\u5728\u5199\u51fa\u7279\u89e3$$y^{*}=x\\frac{1}{F(D)} e^{kx} \\cdot \\varphi (x)$$\u540e\uff0c\u5c06$e^{kx}$\u5f80\u524d\u63d0\uff0c\u5e76\u540c\u65f6\u5c06F(D)\u66ff\u6362\u4e3aF(D+k),\u5982\u4e0b\u56fe\u6240\u793a\uff1a<\/p>\n\n\n\n

\"\"<\/div><\/figure>\n\n\n\n

\u7c7b\u578b\u56db:$f(x)=x\u7684\u591a\u9879\u5f0f$)<\/h2>\n\n\n\n

\u6b64\u7c7b\u578b\u7528\u5f85\u5b9a\u7cfb\u6570\u6cd5\u53ef\u80fd\u4f1a\u66f4\u52a0\u7b80\u5355\uff0c\u8fd9\u91cc\u5c31\u4e0d\u8be6\u7ec6\u8bb2\u89e3\u4e86(busi,\u5bbf\u820d\u65ad\u7535\u4e86\uff0c\u660e\u5929\u518d\u5199QAQ)<\/p>\n","protected":false},"excerpt":{"rendered":"

\u6211\u4eec\u5728\u5c0f\u5b66\u4e00\u5e74\u7ea7\u5c31\u5b66\u8fc7\uff0c\u4e8c\u9636\u5e38\u7cfb\u6570\u975e\u9f50\u6b21\u7ebf\u6027\u5fae\u5206\u65b9\u7a0b\uff1a $$y”+py’+q=f(x) […]<\/p>\n","protected":false},"author":1,"featured_media":14,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[],"class_list":["post-129","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algorithh"],"_links":{"self":[{"href":"https:\/\/qiuqiqaq.fun\/index.php\/wp-json\/wp\/v2\/posts\/129","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/qiuqiqaq.fun\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/qiuqiqaq.fun\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/qiuqiqaq.fun\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/qiuqiqaq.fun\/index.php\/wp-json\/wp\/v2\/comments?post=129"}],"version-history":[{"count":48,"href":"https:\/\/qiuqiqaq.fun\/index.php\/wp-json\/wp\/v2\/posts\/129\/revisions"}],"predecessor-version":[{"id":187,"href":"https:\/\/qiuqiqaq.fun\/index.php\/wp-json\/wp\/v2\/posts\/129\/revisions\/187"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/qiuqiqaq.fun\/index.php\/wp-json\/wp\/v2\/media\/14"}],"wp:attachment":[{"href":"https:\/\/qiuqiqaq.fun\/index.php\/wp-json\/wp\/v2\/media?parent=129"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qiuqiqaq.fun\/index.php\/wp-json\/wp\/v2\/categories?post=129"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qiuqiqaq.fun\/index.php\/wp-json\/wp\/v2\/tags?post=129"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}