{"id":32850,"date":"2022-04-28T13:00:03","date_gmt":"2022-04-28T13:00:03","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32850"},"modified":"2025-04-03T22:55:55","modified_gmt":"2025-04-03T22:55:55","slug":"%e4%bb%80%e4%b9%88%e6%98%af%e5%b8%b8%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%ef%bc%88ode%ef%bc%89%ef%bc%9f","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/zh\/%e4%bb%80%e4%b9%88%e6%98%af%e5%b8%b8%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%ef%bc%88ode%ef%bc%89%ef%bc%9f\/","title":{"rendered":"\u4ec0\u4e48\u662f\u5e38\u5fae\u5206\u65b9\u7a0b\uff08ODE\uff09\uff1f"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>\u4ec0\u4e48\u662f\u5e38\u5fae\u5206\u65b9\u7a0b (ODE)?<\/h1>\n<p style=\"text-align:center;\" dir=\"ltr\"><em><strong>\u6458\u8981\uff1a<\/strong><\/br>\u5728\u672c\u8bfe\u7a0b\u4e2d\uff0c\u6211\u4eec\u63a2\u8ba8\u4e86\u9636\u6570\u4e3a k \u7684\u5e38\u5fae\u5206\u65b9\u7a0b\uff08ODE\uff09\uff0c\u4ece\u5176\u5b9a\u4e49\u4ee5\u53ca\u6807\u51c6\u5f62\u5f0f\u548c\u4e00\u822c\u5f62\u5f0f\u7684\u8868\u793a\u65b9\u6cd5\u5165\u624b\u3002\u901a\u8fc7\u96c5\u53ef\u6bd4\u77e9\u9635\u548c\u9690\u51fd\u6570\u5b9a\u7406\u7b49\u6982\u5ff5\uff0c\u4e3a\u7406\u89e3\u8fd9\u4e9b\u65b9\u7a0b\u7684\u89e3\u53ca\u5176\u76f8\u5173\u6027\u8d28\uff08\u5982\u5b9a\u4e49\u57df\u3001\u663e\u5f0f\u89e3\u4e0e\u9690\u5f0f\u89e3\uff09\u5960\u5b9a\u4e86\u57fa\u7840\u3002<\/em><\/p>\n<p style=\"text-align:center\"><strong>\u5b66\u4e60\u76ee\u6807<\/strong><\/p>\n<p>\u5b8c\u6210\u672c\u8bfe\u7a0b\u540e\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a<\/p>\n<ol>\n<li><strong>\u8bb0\u4f4f<\/strong>\u5e38\u5fae\u5206\u65b9\u7a0b\uff08ODE\uff09\u7684\u5b9a\u4e49\u53ca\u5176\u57fa\u672c\u7279\u5f81\u3002<\/li>\n<li><strong>\u89e3\u91ca<\/strong>ODE \u4e0e\u5176\u53ef\u80fd\u89e3\u4e4b\u95f4\u7684\u5173\u7cfb\u3002<\/li>\n<\/ol>\n<p style=\"text-align:center;\" dir=\"ltr\"><strong>\u76ee\u5f55<\/strong><br \/>\n<a href=\"#LaEcuacionDiferencialOrdinariaDeOrdenK\"><strong>\u9636\u6570\u4e3a k \u7684\u5e38\u5fae\u5206\u65b9\u7a0b (ODE)<\/strong><\/a><br \/>\n<a href=\"#TeoremaDeLaFuncionImplicita\">\u9690\u51fd\u6570\u5b9a\u7406<\/a><br \/>\n<a href=\"#LaSolucionDeUnaEcuacionDiferencialOrdinaria\"><strong>\u5e38\u5fae\u5206\u65b9\u7a0b\u7684\u89e3<\/strong><\/a><br \/>\n<a href=\"#CuidadoConElDominioDeDefinicionDeLasSoluciones\">\u6ce8\u610f\u89e3\u7684\u5b9a\u4e49\u57df<\/a><br \/>\n<a href=\"#SolucionExtendidaYSolucionMaximal\">\u62d3\u5c55\u89e3\u4e0e\u6700\u5927\u89e3<\/a><br \/>\n<a href=\"#SolucionExplicitaYSolucionImplicita\">\u663e\u5f0f\u89e3\u4e0e\u9690\u5f0f\u89e3<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/zE29azRIKng\" title=\"YouTube \u89c6\u9891\u64ad\u653e\u5668\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p>\u901a\u8fc7\u76ee\u524d\u7684\u5b66\u4e60\uff0c\u6211\u4eec\u5df2\u7ecf\u5bf9\u5fae\u5206\u65b9\u7a0b\u7684\u6982\u5ff5\u4ee5\u53ca\u5b83\u4eec\u53ef\u80fd\u7684\u591a\u79cd\u5e94\u7528\u6709\u4e86\u76f8\u5f53\u6e05\u6670\u7684\u7406\u89e3\u3002\u73b0\u5728\u6211\u4eec\u5c06\u6682\u505c\u4e0b\u6765\uff0c\u7814\u7a76\u4e00\u4e9b\u5b9a\u4e49\u4e0e\u6027\u8d28\uff0c\u4ee5\u5efa\u7acb\u4e00\u4e2a\u575a\u5b9e\u7684\u5171\u540c\u57fa\u7840\u6765\u7ee7\u7eed\u6df1\u5165\u5b66\u4e60\u3002<\/p>\n<p><a name=\"LaEcuacionDiferencialOrdinariaDeOrdenK\"><\/a><\/p>\n<h3>\u9636\u6570\u4e3a k \u7684\u5e38\u5fae\u5206\u65b9\u7a0b<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=163s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u5e38\u5fae\u5206\u65b9\u7a0b\uff08ODE\uff09<\/span><\/strong><\/a> \u662f\u4e00\u79cd\u5305\u542b\u4e00\u4e2a\u81ea\u53d8\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span>\u3001\u4e00\u4e2a\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span><\/span> \u4ee5\u53ca\u8be5\u51fd\u6570\u67d0\u4e9b\u9636\u6570\u7684\u666e\u901a\u5bfc\u6570\u7684\u65b9\u7a0b\u3002<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y(x)<\/span><\/span> \u7684\u4e00\u9636\u5bfc\u6570\u53ef\u4ee5\u8bb0\u4f5c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{dy(x)}{dx}<\/span><\/span> \u6216 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y&#039;(x)<\/span><\/span>\uff0c\u4e8c\u9636\u5bfc\u6570\u53ef\u4ee5\u8bb0\u4f5c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d^2y(x)}{dx^2}<\/span><\/span> \u6216 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y&#039;&#039;(x)<\/span><\/span>\uff0c\u4e00\u822c\u5730\uff0cn \u9636\u5bfc\u6570\u8bb0\u4f5c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d^ny(x)}{dx^n}<\/span><\/span> \u6216 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(n)}(x)<\/span><\/span>\u3002\u82e5\u67d0\u4e2a\u6700\u5927\u503c <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(k)}(x)<\/span><\/span> \u51fa\u73b0\u5728\u65b9\u7a0b\u4e2d\uff0c\u5219\u79f0\u8be5\u65b9\u7a0b\u7684<strong>\u9636\u6570\u4e3a k<\/strong>\u3002\u56e0\u6b64\uff0c\u9636\u6570\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u7684\u5e38\u5fae\u5206\u65b9\u7a0b\u7684<strong>\u4e00\u822c\u5f62\u5f0f<\/strong>\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F\\left(x,y(x),y&#039;(x), \\cdots, y^{(k)}(x)\\right)=0.<\/span>\n<p>\u5f53\u8be5\u65b9\u7a0b\u53ef\u4ee5\u5c06 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(k)}(x)<\/span><\/span> \u663e\u5f0f\u8868\u793a\u51fa\u6765\u65f6\uff0c\u6211\u4eec\u8bf4\u8be5\u9636\u6570\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u7684\u5e38\u5fae\u5206\u65b9\u7a0b\u662f<strong>\u6807\u51c6\u5f62\u5f0f<\/strong>\uff0c\u5373\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(k)}(x) = f\\left(x,y(x),y&#039;(x), \\cdots, y^{(k-1)}(x)\\right).<\/span>\n<p>\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">y<\/span> \u662f\u4e00\u4e2a\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R} \\longrightarrow \\mathbb{R}^n,<\/span><\/span>\uff0c\u610f\u5473\u7740\u5b83\u53ca\u5176\u5bfc\u6570\u5728\u4efb\u610f\u4e00\u70b9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in\\mathbb{R}<\/span><\/span> \u4e0a\u7684\u503c\u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> \u4e2d\u7684\u5411\u91cf\u3002\u5728\u6b64\u524d\u63d0\u4e0b\uff0c\u6211\u4eec\u5f97\u51fa\uff1a\u63cf\u8ff0\u9636\u6570\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u7684\u5e38\u5fae\u5206\u65b9\u7a0b\u7684\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> \u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1+(k+1)<\/span><\/span> \u4e2a\u53d8\u91cf\uff0c\u56e0\u6b64 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Dom}(F)\\subset \\mathbb{R}^{1+n(k+1)}<\/span><\/span>\uff0c\u800c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Rec}(F)\\subset \\mathbb{R}<\/span><\/span>\uff1b\u7c7b\u4f3c\u5730\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Dom}(f) = \\mathbb{R}^{1+nk}<\/span><\/span> \u4e14 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Rec}(f)\\subset \\mathbb{R}^n<\/span><\/span>\u3002<\/p>\n<p>\u4ece\u9636\u6570\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u7684\u5e38\u5fae\u5206\u65b9\u7a0b\u7684\u4e00\u822c\u5f62\u5f0f\u8f6c\u6362\u4e3a\u6807\u51c6\u5f62\u5f0f\uff0c\u662f\u4f9d\u8d56\u4e8e<strong>\u9690\u51fd\u6570\u5b9a\u7406<\/strong>\u7684\u3002<\/p>\n<p><a name=\"TeoremaDeLaFuncionImplicita\"><\/a><\/p>\n<h4>\u9690\u51fd\u6570\u5b9a\u7406<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=887s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u8bbe <span class=\"katex-eq\" data-katex-display=\"false\">F<\/span> \u662f\u5b9a\u4e49\u5728\u5f00\u96c6 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">U \\subset \\mathbb{R}^n<\/span><\/span> \u4e0a\u7684 <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{C}^1<\/span> \u7c7b\u51fd\u6570<\/span><\/strong><\/a>\uff0c\u5176\u53d6\u503c\u4e3a\u5b9e\u6570\u3002\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1,\\cdots, a_n) \\in U<\/span><\/span>\uff0c\u4e14\u6ee1\u8db3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(a_1,\\cdots, a_n) = 0<\/span><\/span> \u4e14<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial F(a_1,\\cdots, a_n)}{\\partial x_n} \\neq 0<\/span>\n<p>\u90a3\u4e48\u5b58\u5728\u4e00\u4e2a\u90bb\u57df <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span>\uff0c\u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1, \\cdots, a_{n-1}) \\in \\mathbb{R}^{n-1}<\/span><\/span>\uff0c\u5e76\u5b58\u5728\u4e00\u4e2a\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi:V \\longrightarrow \\mathbb{R}<\/span><\/span> \u4f7f\u5f97\uff1a<\/p>\n<ol>\n<li type=\"i\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V \\times \\varphi(V) \\subset U<\/span><\/span><\/li>\n<li type=\"i\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(x_1,\\cdots,x_{n-1},x_n) = 0 \\leftrightarrow x_n = \\varphi(x_1,\\cdots, x_{n-1})<\/span><\/span><\/li>\n<li type=\"i\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi<\/span> \u53ef\u5fae\uff0c\u4e14\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\dfrac{\\partial \\varphi (a_1,\\cdots, a_{n-1})}{\\partial x_i} = - \\dfrac{ \\dfrac{\\partial F (a_1,\\cdots, a_n)}{\\partial x_i} }{ \\dfrac{\\partial F (a_1,\\cdots, a_n)}{\\partial x_n} }<\/span>\n<\/li>\n<\/ol>\n<h4>\u9690\u51fd\u6570\u5b9a\u7406\u7684\u8bc1\u660e<\/h4>\n<h5>\u4ece\u96c5\u53ef\u6bd4\u77e9\u9635\u51fa\u53d1\u7684\u63a8\u5bfc<\/h5>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=1101s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi(x_1,\\cdots,x_{n-1}, x_n) = (x_1,\\cdots,x_{n-1}, F(x_1,\\cdots, x_n)).<\/span><\/span><\/span><\/strong><\/a> \u82e5\u6211\u4eec\u8ba1\u7b97\u5176\u96c5\u53ef\u6bd4\u77e9\u9635\uff0c\u5982\u4e0b\u6240\u793a\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left( \\dfrac{\\partial \\psi(x_1,\\cdots, x_n)}{\\partial(x_1,\\cdots, x_n)} \\right) = \\left( \\begin{array}{cccc}\n\n1 &amp; 0 &amp;  \\cdots &amp; 0 \\\\\n\n0 &amp; 1 &amp;  \\cdots &amp; \\vdots \\\\\n\n\\vdots &amp;\\vdots &amp; \\ddots  &amp; \\vdots  \\\\\n\n\\displaystyle \\dfrac{\\partial F(x_1, \\cdots, x_n)}{\\partial x_1} &amp; \\dfrac{\\partial F(x_1, \\cdots, x_n)}{\\partial x_2} &amp; \\cdots  &amp; \\dfrac{\\partial F(x_1, \\cdots, x_n)}{\\partial x_n}\n\n\\end{array}\\right), <\/span>\n<p>\u6211\u4eec\u53ef\u4ee5\u770b\u5230\u8be5\u77e9\u9635\u5728\u70b9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1,\\cdots, a_n)<\/span><\/span> \u5904\u7684\u884c\u5217\u5f0f\u4e0d\u4e3a\u96f6\uff0c\u6b63\u5982\u6211\u4eec\u4e00\u5f00\u59cb\u6240\u8bbe <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\partial F(a_1,\\cdots, a_n)\/\\partial x_n \\neq 0.<\/span><\/span> \u57fa\u4e8e\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> \u5728\u5305\u542b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1,\\cdots, a_n)<\/span><\/span> \u7684\u5f00\u96c6 <span class=\"katex-eq\" data-katex-display=\"false\">W<\/span> \u4e0a\u662f\u53ef\u9006\u7684\u3002<\/p>\n<h5>\u89e3\u7684\u6784\u9020<\/h5>\n<p>\u73b0\u5728\uff0c\u8003\u8651\u96c6\u5408<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{V}=\\psi(W)\\ni \\psi(a_1,\\cdots,a_{n}) = (a_1,\\cdots,a_{n-1},F(a_1,\\cdots,a_{n}))=(a_1,\\cdots,a_{n-1},0).<\/span>\n<p>\u636e\u6b64\u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\u53e6\u4e00\u4e2a\u96c6\u5408<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V=\\{(x_1,\\cdots,x_{n-1}) \\;|\\; (x_1,\\cdots,x_{n-1},0)\\in \\tilde{V}\\}\\ni (a_1,\\cdots,a_{n-1})<\/span>\n<p>\u96c6\u5408 <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> \u56e0\u6b64\u662f\u4e00\u4e2a\u5305\u542b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a_1,\\cdots,a_{n-1})\\in\\mathbb{R}^{n-1}<\/span><\/span> \u7684\u5f00\u96c6\u3002<\/p>\n<p>\u6b64\u5916\uff0c\u7531\u4e8e <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> \u5728 <span class=\"katex-eq\" data-katex-display=\"false\">W<\/span> \u4e0a\u53ef\u9006\uff0c\u5b58\u5728\u552f\u4e00\u7684 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(y_1,\\cdots,y_n)\\in W<\/span><\/span> \u4f7f\u5f97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi(y_1,\\cdots,y_n) = (x_1,\\cdots,x_{n-1},0)<\/span><\/span>\u3002\u8fd9\u610f\u5473\u7740\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} y_1 &amp;= x_1 \\\\ \\\\ \\vdots &amp; \\vdots \\\\ \\\\ y_{n-1} &amp;= x_{n-1} \\\\ \\\\ F(x_1,\\cdots,x_{n-1},y_n) &amp;= 0 \\end{array}<\/span>\n<p>\u7531\u6b64\u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi(x_1,\\cdots,x_{n-1}) = y_n<\/span><\/span>\uff0c\u4f7f\u5f97\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi^{-1}(x_1,\\cdots,x_{n-1},0) = (x_1,\\cdots,x_{n-1},\\varphi(x_1,\\cdots,x_{n-1}))<\/span>\n<p>\u4e14<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(x_1,\\cdots,x_{n-1},\\varphi(x_1,\\cdots,x_{n-1})) = 0<\/span>\n<p>\u636e\u6b64\u53ef\u77e5\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi(V)\\ni a_n,<\/span><\/span> \u4ece\u800c\u6709 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">V\\times\\varphi(V) \\subset U,<\/span><\/span> \u5e76\u4e14\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(x_1,\\cdots,x_{n-1},x_n) = 0 \\leftrightarrow x_n = \\varphi(x_1,\\cdots,x_{n-1})<\/span>\n<h5>\u53ef\u5fae\u6027<\/h5>\n<p>\u6700\u540e\uff0c<span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> \u7684\u53ef\u5fae\u6027\u610f\u5473\u7740\u5176\u9006\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi^{-1}<\/span><\/span> \u4e5f\u53ef\u5fae\uff0c\u4ece\u800c\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">\\varphi<\/span> \u5728 <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> \u4e0a\u53ef\u5fae\u3002\u5728\u6b64\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u5b9a\u4e49\u4e00\u4e2a\u51fd\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span> \u5982\u4e0b\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g(x_1, \\cdots,x_{n-1}) = F(x_1,\\cdots,x_{n-1},\\varphi(x_1,\\cdots,x_{n-1})) = 0<\/span>\n<p>\u63a5\u7740\uff0c\u5229\u7528\u94fe\u5f0f\u6cd5\u5219\u6211\u4eec\u6709\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial g}{\\partial x_i} = \\frac{\\partial F}{\\partial x_i} + \\frac{\\partial F}{\\partial x_n}\\frac{\\partial \\varphi }{\\partial x_i} = 0,<\/span>\n<p>\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i=1,\\cdots, n-1.<\/span><\/span> \u7531\u6b64\u53ef\u5f97\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\dfrac{\\partial \\varphi(a_1,\\cdots,a_{n-1})}{\\partial x_i} = - \\dfrac{\\dfrac{\\partial F(a_1,\\cdots,a_{n})}{\\partial x_i}}{\\dfrac{\\partial F(a_1,\\cdots,a_{n})}{\\partial x_n}}<\/span>\n<p>\u81f3\u6b64\uff0c\u6240\u6709\u9700\u8981\u8bc1\u660e\u7684\u5185\u5bb9\u5df2\u5b8c\u6210 \u25a0<\/p>\n<p><a name=\"LaSolucionDeUnaEcuacionDiferencialOrdinaria\"><\/a><\/p>\n<h3>\u5e38\u5fae\u5206\u65b9\u7a0b\u7684\u89e3<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=2249s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u6211\u4eec\u8003\u8651\u4ee5\u6807\u51c6\u5f62\u5f0f\u8868\u8fbe\u7684\u5e38\u5fae\u5206\u65b9\u7a0b<\/span><\/strong><\/a><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(n)} = f(x,y(x),y^\\prime(x),\\cdots,y^{(n-1)(x)})<\/span>\n<p>\u90a3\u4e48\uff0c\u4e00\u4e2a\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\varphi : I_\\phi \\longmapsto \\mathbb{R}^n,<\/span><\/span>\uff0c\u5176\u4e2d <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_\\phi<\/span><\/span> \u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span><\/span> \u4e2d\u7684\u4e00\u4e2a\u533a\u95f4\uff0c\u5982\u679c\u6ee1\u8db3\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall x \\in I_\\phi \\right) \\left(\\varphi^{(n)}(x) = f(x,\\varphi(x),\\varphi^\\prime(x),\\cdots,\\varphi^{(n-1)(x)}\\right)<\/span>\n<p>\u5219\u79f0 <strong>\u51fd\u6570\u662f\u8be5\u5e38\u5fae\u5206\u65b9\u7a0b\u7684\u4e00\u4e2a\u89e3<\/strong>\u3002<\/p>\n<p><a name=\"CuidadoConElDominioDeDefinicionDeLasSoluciones\"><\/a><\/p>\n<h4>\u6ce8\u610f\u89e3\u7684\u5b9a\u4e49\u57df<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=2387s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u5728\u8fd9\u91cc\u5fc5\u987b\u7279\u522b\u5f3a\u8c03<\/span><\/strong><\/a> \u660e\u786e\u58f0\u660e\u5fae\u5206\u65b9\u7a0b\u89e3\u7684\u5b9a\u4e49\u57df\u7684\u91cd\u8981\u6027\u3002\u4f8b\u5982\uff0c\u5728\u524d\u4e00\u6bb5\u4e2d\u63d0\u5230\u7684\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span><\/span> \u7684\u5b9a\u4e49\u57df\u4e3a\u533a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_\\phi.<\/span><\/span> \u8fd9\u662f\u91cd\u8981\u7684\uff0c\u56e0\u4e3a\u5728\u5904\u7406\u5e38\u5fae\u5206\u65b9\u7a0b\u65f6\uff0c\u4e00\u4e2a\u5e38\u89c1\u9519\u8bef\u662f\u5c06\u4e24\u4e2a\u89e3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_1<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_2<\/span><\/span> \u89c6\u4e3a\u76f8\u540c\uff0c\u4ec5\u56e0\u4e3a\u5b83\u4eec\u5728\u516c\u5171\u533a\u95f4\u5185\u7684\u51fd\u6570\u503c\u76f8\u7b49\uff0c\u5373\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall x \\in I_{\\phi_1}\\cap I_{\\phi_2}\\right)\\left(\\phi_1(x) = \\phi_2(x)\\right),<\/span>\n<p>\u5c3d\u7ba1 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{\\phi_1}\\neq I_{\\phi_2}.<\/span><\/span> \u4e3a\u4e86\u8bf4\u660e\u8fd9\u4e00\u70b9\uff0c\u6211\u4eec\u6765\u770b\u5982\u4e0b\u7684\u5fae\u5206\u65b9\u7a0b\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^\\prime = -y^2.<\/span>\n<p>\u8be5\u65b9\u7a0b\u7684\u4e00\u4e2a\u53ef\u80fd\u89e3\u662f\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1 : ]0,+\\infty[ \\longrightarrow \\mathbb{R}^+\\setminus\\{0\\}<\/span><\/span>\uff0c\u5176\u5b9a\u4e49\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1(x)=1\/x,<\/span><\/span>\uff0c\u56e0\u4e3a\u5bf9\u4efb\u610f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]0,+\\infty[<\/span><\/span> \u90fd\u6709\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1^{\\prime} = -1\/x^2 = -\\psi_1^2<\/span>\n<p>\u4f46\u5982\u679c\u6211\u4eec\u4e0d\u6ce8\u610f\u7ec6\u8282\uff0c\u901a\u8fc7\u4ee3\u6570\u6280\u5de7\uff0c\u53ef\u4ee5\u6784\u9020\u51fa\u4e00\u4e2a\u5b8c\u5168\u4e0d\u540c\u7684\u89e3\u3002\u4f8b\u5982\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{1}{x} = \\frac{1}{1 - (1-x)},<\/span>\n<p>\u8be5\u7b49\u5f0f\u53f3\u8fb9\u5b9e\u9645\u4e0a\u662f\u51e0\u4f55\u7ea7\u6570\u7684\u548c\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\displaystyle \\sum_{n=0}^{+\\infty} (1-x)^n = \\frac{1}{1 - (1-x)}<\/span>\n<p>\u56e0\u6b64\uff0c\u5bf9\u4e8e\u4e0d\u719f\u6089\u8fd9\u4e9b\u6280\u5de7\u7684\u4eba\u6765\u8bf4\uff0c\u53ef\u80fd\u4f1a\u9519\u8bef\u5730\u8ba4\u4e3a\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1<\/span><\/span> \u4e0e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_2 = \\sum_{n=0}^{+\\infty} (1-x)^n <\/span><\/span> \u63d0\u4f9b\u4e86\u76f8\u540c\u7684\u5fae\u5206\u65b9\u7a0b\u89e3\uff0c\u56e0\u4e3a\u5b83\u4eec\u5728\u67d0\u4e9b\u70b9\u7684\u51fd\u6570\u503c\u786e\u5b9e\u76f8\u540c\u3002\u4f46\u4f1a\u5ffd\u7565\u4e00\u4e2a\u5173\u952e\u70b9\uff1a\u8be5\u51e0\u4f55\u7ea7\u6570\u4ec5\u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|1-x| \\lt 1<\/span><\/span>\uff0c\u4e5f\u5c31\u662f\u5f53 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\in]0,2[<\/span><\/span> \u65f6\u624d\u6536\u655b\u3002\u800c\u4e14\uff0c\u65e2\u7136 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">]0,2[\\subset]0,+\\infty[<\/span><\/span>\uff0c\u6211\u4eec\u6709\u7406\u7531\u8bf4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1<\/span><\/span> \u662f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_2<\/span><\/span> \u7684\u5ef6\u62d3\u51fd\u6570\uff0c\u56e0\u4e3a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_1<\/span><\/span> \u5728 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\psi_2<\/span><\/span> \u6709\u5b9a\u4e49\u7684\u5730\u65b9\u4e5f\u6709\u5b9a\u4e49\uff0c\u751a\u81f3\u5728\u66f4\u5927\u7684\u533a\u95f4\u4e0a\u4e5f\u6709\u5b9a\u4e49\u3002<\/p>\n<p><a name=\"SolucionExtendidaYSolucionMaximal\"><\/a><\/p>\n<h4>\u62d3\u5c55\u89e3\u4e0e\u6700\u5927\u89e3<\/h4>\n<p>\u6211\u4eec\u8003\u8651\u4e24\u4e2a\u51fd\u6570 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_1<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_2<\/span><\/span>\uff0c\u5206\u522b\u5b9a\u4e49\u5728\u533a\u95f4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{\\phi_1}<\/span><\/span> \u548c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{\\phi_2}<\/span><\/span> \u4e0a\uff0c\u5b83\u4eec\u90fd\u662f\u67d0\u4e2a\u5fae\u5206\u65b9\u7a0b\u7684\u89e3\u3002\u5982\u679c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I_{\\phi_1}\\subset I_{\\phi_2},<\/span><\/span> \u90a3\u4e48\u6211\u4eec\u8bf4\u89e3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_2<\/span><\/span> \u662f\u5bf9\u89e3 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_1<\/span><\/span> \u7684<strong>\u62d3\u5c55<\/strong>\uff0c\u6216\u8005\u8bf4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_2<\/span><\/span> \u662f\u6bd4 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_1<\/span><\/span> \u66f4\u4e00\u822c\u7684\u89e3\u3002\u5982\u679c\u4e00\u4e2a\u89e3 <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> \u4e0d\u5b58\u5728\u5176\u4ed6\u975e\u5e73\u51e1\u65b9\u5f0f\u53ef\u4ee5\u5c06\u5176\u62d3\u5c55\u7684\u60c5\u5f62\uff0c\u5219\u79f0\u5b83\u4e3a\u201c\u6700\u5927\u89e3\u201d\u3002<\/p>\n<p><a name=\"SolucionExplicitaYSolucionImplicita\"><\/a><\/p>\n<h4>\u663e\u5f0f\u89e3\u4e0e\u9690\u5f0f\u89e3<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=zE29azRIKng&#038;t=2649s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">\u4e00\u4e2a\u51fd\u6570<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> \u88ab\u8ba4\u4e3a\u662f\u9636\u6570\u4e3a <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u7684\u5e38\u5fae\u5206\u65b9\u7a0b\uff08\u4ee5\u6807\u51c6\u5f62\u5f0f\u8868\u793a\uff09\u7684\u89e3\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^{(n)}(x)=f(x,y(x),y^\\prime(x),\\cdots,y^{(n-1)}(x)),<\/span>\n<p>\u82e5\u5728\u533a\u95f4 <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> \u4e0a\u6ee1\u8db3\uff1a<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall x\\in I)\\left(\\phi^{n}(x) = f(x,\\phi(x),\\phi^\\prime(x),\\cdots,\\phi^{(n-1)}(x))\\right)<\/span>\n<p>\u6211\u4eec\u5728\u524d\u9762\u51e0\u6bb5\u6240\u8ba8\u8bba\u7684\u5c31\u662f\u6240\u8c13\u7684<strong>\u8be5\u5fae\u5206\u65b9\u7a0b\u5728\u533a\u95f4 <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> \u4e0a\u7684\u663e\u5f0f\u89e3<\/strong>\u3002\u6b63\u5982\u540d\u79f0\u6240\u793a\uff0c\u4e5f\u5b58\u5728\u4ee5\u9690\u5f0f\u65b9\u5f0f\u5b9a\u4e49\u89e3\u7684\u5f62\u5f0f\u3002\u6211\u4eec\u79f0\u4e00\u4e2a\u5173\u7cfb\u5f0f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi(x,y)=0<\/span><\/span> \u4e3a<strong>\u8be5\u5fae\u5206\u65b9\u7a0b\u5728\u533a\u95f4 <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> \u4e0a\u7684\u9690\u5f0f\u89e3<\/strong>\uff0c\u5982\u679c\u5b83\u5728 <span class=\"katex-eq\" data-katex-display=\"false\">I<\/span> \u4e0a\u5b9a\u4e49\u4e86\u4e24\u4e2a\u6216\u591a\u4e2a\u9690\u5f0f\u89e3\u3002<\/p>\n<h3>\u7ed3\u8bed<\/h3>\n<p>\u5728\u672c\u8282\u8bfe\u4e2d\uff0c\u6211\u4eec\u4ee5\u4e25\u8c28\u800c\u6613\u61c2\u7684\u65b9\u5f0f\u5256\u6790\u4e86\u5e38\u5fae\u5206\u65b9\u7a0b\u7684\u6982\u5ff5\uff0c\u5efa\u7acb\u4e86\u5f62\u5f0f\u5316\u7684\u57fa\u7840\uff0c\u4f7f\u6211\u4eec\u4e0d\u4ec5\u80fd\u591f\u8bc6\u522b\u4e00\u4e2a\u5e38\u5fae\u5206\u65b9\u7a0b\uff0c\u8fd8\u80fd\u7406\u89e3\u5176\u89e3\u7684\u80cc\u540e\u903b\u8f91\u3002\u591a\u4e8f\u4e86\u9690\u51fd\u6570\u5b9a\u7406\uff0c\u6211\u4eec\u5f97\u4ee5\u6e05\u695a\u5730\u89e3\u91ca\u4ece\u5176\u4e00\u822c\u5f62\u5f0f\u5230\u6807\u51c6\u5f62\u5f0f\u7684\u8f6c\u6362\uff0c\u8fd9\u5bf9\u4e8e\u5904\u7406\u5177\u4f53\u95ee\u9898\u662f\u4e00\u9879\u5173\u952e\u7684\u6280\u672f\u80fd\u529b\u3002<\/p>\n<p>\u6b64\u5916\uff0c\u6211\u4eec\u6e05\u695a\u5730\u533a\u5206\u4e86\u89e3\u7684\u4e0d\u540c\u7c7b\u578b\uff1a\u663e\u5f0f\u89e3\u4e0e\u9690\u5f0f\u89e3\uff0c\u62d3\u5c55\u89e3\u4e0e\u6700\u5927\u89e3\uff0c\u5e76\u7279\u522b\u5f3a\u8c03\u4e86\u5bf9\u5b9a\u4e49\u57df\u4f5c\u51fa\u660e\u786e\u58f0\u660e\u7684\u91cd\u8981\u6027\u2014\u2014\u8fd9\u4e00\u70b9\u5e38\u5e38\u88ab\u4f4e\u4f30\u3002\u8fd9\u4e9b\u533a\u522b\u4e0d\u4ec5\u662f\u5f62\u5f0f\u4e0a\u7684\uff0c\u66f4\u662f\u64cd\u4f5c\u4e0a\u7684\uff1b\u5ffd\u89c6\u5b83\u4eec\uff0c\u6b63\u5982\u6211\u4eec\u6240\u89c1\uff0c\u53ef\u80fd\u5bfc\u81f4\u4e25\u91cd\u7684\u6982\u5ff5\u6027\u9519\u8bef\u3002<\/p>\n<p>\u901a\u8fc7\u8fd9\u4e2a\u7ed3\u5c3e\uff0c\u6211\u4eec\u5df2\u83b7\u5f97\u4e86\u4e00\u628a\u950b\u5229\u7684\u5de5\u5177\u3002\u7406\u89e3\u5e38\u5fae\u5206\u65b9\u7a0b\u4e0d\u5e94\u4ec5\u4ec5\u505c\u7559\u5728\u6c42\u89e3\u4e00\u4e2a\u516c\u5f0f\u4e0a\uff1a\u5b83\u9700\u8981\u6279\u5224\u6027\u601d\u7ef4\u3001\u5bf9\u7ec6\u8282\u7684\u5173\u6ce8\uff0c\u4ee5\u53ca\u7262\u56fa\u7684\u6982\u5ff5\u57fa\u7840\uff0c\u4ece\u800c\u4f7f\u6211\u4eec\u80fd\u6301\u7eed\u524d\u8fdb\uff0c\u4e0d\u8ff7\u5931\u65b9\u5411\u3002\u800c\u8fd9\uff0c\u4ec5\u4ec5\u662f\u4e2a\u5f00\u59cb\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4ec0\u4e48\u662f\u5e38\u5fae\u5206\u65b9\u7a0b (ODE)? \u6458\u8981\uff1a\u5728\u672c\u8bfe\u7a0b\u4e2d\uff0c\u6211\u4eec\u63a2\u8ba8\u4e86\u9636\u6570\u4e3a k \u7684\u5e38\u5fae\u5206\u65b9\u7a0b\uff08ODE\uff09\uff0c\u4ece\u5176\u5b9a\u4e49\u4ee5\u53ca\u6807\u51c6\u5f62\u5f0f\u548c\u4e00\u822c\u5f62\u5f0f\u7684\u8868\u793a\u65b9\u6cd5\u5165\u624b\u3002\u901a\u8fc7\u96c5\u53ef\u6bd4\u77e9\u9635\u548c\u9690\u51fd\u6570\u5b9a\u7406\u7b49\u6982\u5ff5\uff0c\u4e3a\u7406\u89e3\u8fd9\u4e9b\u65b9\u7a0b\u7684\u89e3\u53ca\u5176\u76f8\u5173\u6027\u8d28\uff08\u5982\u5b9a\u4e49\u57df\u3001\u663e\u5f0f\u89e3\u4e0e\u9690\u5f0f\u89e3\uff09\u5960\u5b9a\u4e86\u57fa\u7840\u3002 \u5b66\u4e60\u76ee\u6807 \u5b8c\u6210\u672c\u8bfe\u7a0b\u540e\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a \u8bb0\u4f4f\u5e38\u5fae\u5206\u65b9\u7a0b\uff08ODE\uff09\u7684\u5b9a\u4e49\u53ca\u5176\u57fa\u672c\u7279\u5f81\u3002 \u89e3\u91caODE \u4e0e\u5176\u53ef\u80fd\u89e3\u4e4b\u95f4\u7684\u5173\u7cfb\u3002 \u76ee\u5f55 \u9636\u6570\u4e3a k \u7684\u5e38\u5fae\u5206\u65b9\u7a0b (ODE) \u9690\u51fd\u6570\u5b9a\u7406 \u5e38\u5fae\u5206\u65b9\u7a0b\u7684\u89e3 \u6ce8\u610f\u89e3\u7684\u5b9a\u4e49\u57df \u62d3\u5c55\u89e3\u4e0e\u6700\u5927\u89e3 \u663e\u5f0f\u89e3\u4e0e\u9690\u5f0f\u89e3 \u901a\u8fc7\u76ee\u524d\u7684\u5b66\u4e60\uff0c\u6211\u4eec\u5df2\u7ecf\u5bf9\u5fae\u5206\u65b9\u7a0b\u7684\u6982\u5ff5\u4ee5\u53ca\u5b83\u4eec\u53ef\u80fd\u7684\u591a\u79cd\u5e94\u7528\u6709\u4e86\u76f8\u5f53\u6e05\u6670\u7684\u7406\u89e3\u3002\u73b0\u5728\u6211\u4eec\u5c06\u6682\u505c\u4e0b\u6765\uff0c\u7814\u7a76\u4e00\u4e9b\u5b9a\u4e49\u4e0e\u6027\u8d28\uff0c\u4ee5\u5efa\u7acb\u4e00\u4e2a\u575a\u5b9e\u7684\u5171\u540c\u57fa\u7840\u6765\u7ee7\u7eed\u6df1\u5165\u5b66\u4e60\u3002 \u9636\u6570\u4e3a k \u7684\u5e38\u5fae\u5206\u65b9\u7a0b \u5e38\u5fae\u5206\u65b9\u7a0b\uff08ODE\uff09 \u662f\u4e00\u79cd\u5305\u542b\u4e00\u4e2a\u81ea\u53d8\u91cf \u3001\u4e00\u4e2a\u51fd\u6570 \u4ee5\u53ca\u8be5\u51fd\u6570\u67d0\u4e9b\u9636\u6570\u7684\u666e\u901a\u5bfc\u6570\u7684\u65b9\u7a0b\u3002 \u7684\u4e00\u9636\u5bfc\u6570\u53ef\u4ee5\u8bb0\u4f5c \u6216 \uff0c\u4e8c\u9636\u5bfc\u6570\u53ef\u4ee5\u8bb0\u4f5c \u6216 \uff0c\u4e00\u822c\u5730\uff0cn \u9636\u5bfc\u6570\u8bb0\u4f5c \u6216 \u3002\u82e5\u67d0\u4e2a\u6700\u5927\u503c \u4f7f\u5f97 \u51fa\u73b0\u5728\u65b9\u7a0b\u4e2d\uff0c\u5219\u79f0\u8be5\u65b9\u7a0b\u7684\u9636\u6570\u4e3a k\u3002\u56e0\u6b64\uff0c\u9636\u6570\u4e3a \u7684\u5e38\u5fae\u5206\u65b9\u7a0b\u7684\u4e00\u822c\u5f62\u5f0f\u4e3a\uff1a \u5f53\u8be5\u65b9\u7a0b\u53ef\u4ee5\u5c06 \u663e\u5f0f\u8868\u793a\u51fa\u6765\u65f6\uff0c\u6211\u4eec\u8bf4\u8be5\u9636\u6570\u4e3a \u7684\u5e38\u5fae\u5206\u65b9\u7a0b\u662f\u6807\u51c6\u5f62\u5f0f\uff0c\u5373\uff1a \u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u51fd\u6570 \u662f\u4e00\u4e2a\u51fd\u6570 \uff0c\u610f\u5473\u7740\u5b83\u53ca\u5176\u5bfc\u6570\u5728\u4efb\u610f\u4e00\u70b9 \u4e0a\u7684\u503c\u662f 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