{"id":10802,"date":"2023-08-10T20:28:18","date_gmt":"2023-08-10T12:28:18","guid":{"rendered":"https:\/\/blog.iyatt.com\/?p=10802"},"modified":"2024-05-06T08:00:02","modified_gmt":"2024-05-06T00:00:02","slug":"%e9%ab%98%e9%98%b6%e5%af%bc%e6%95%b0%e6%b1%82%e5%af%bc","status":"publish","type":"post","link":"https:\/\/blog.iyatt.com\/?p=10802","title":{"rendered":"\u9ad8\u9636\u5bfc\u6570\u6c42\u5bfc"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 ez-toc-wrap-center counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">\u76ee\u5f55<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/blog.iyatt.com\/?p=10802\/#%E5%AE%9A%E4%B9%89\" >\u5b9a\u4e49<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/blog.iyatt.com\/?p=10802\/#%E6%B1%82%E5%AF%BC%E8%AE%A1%E7%AE%97\" >\u6c42\u5bfc\u8ba1\u7b97<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/blog.iyatt.com\/?p=10802\/#%E7%9B%B4%E6%8E%A5%E6%B3%95\" >\u76f4\u63a5\u6cd5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/blog.iyatt.com\/?p=10802\/#%E9%97%B4%E6%8E%A5%E6%B3%95\" >\u95f4\u63a5\u6cd5<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/blog.iyatt.com\/?p=10802\/#%E5%9B%9B%E5%88%99%E8%BF%90%E7%AE%97%E6%B3%95%E5%88%99\" >\u56db\u5219\u8fd0\u7b97\u6cd5\u5219<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/blog.iyatt.com\/?p=10802\/#%E5%B8%B8%E7%94%A8%E9%AB%98%E9%98%B6%E5%AF%BC%E6%95%B0%E5%85%AC%E5%BC%8F\" >\u5e38\u7528\u9ad8\u9636\u5bfc\u6570\u516c\u5f0f<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/blog.iyatt.com\/?p=10802\/#%E9%83%A8%E5%88%86%E5%87%BD%E6%95%B0%E4%BA%8C%E9%98%B6%E5%AF%BC%E6%95%B0%E6%B1%82%E6%B3%95\" >\u90e8\u5206\u51fd\u6570\u4e8c\u9636\u5bfc\u6570\u6c42\u6cd5<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/blog.iyatt.com\/?p=10802\/#%E6%8A%BD%E8%B1%A1%E5%A4%8D%E5%90%88%E5%87%BD%E6%95%B0\" >\u62bd\u8c61\u590d\u5408\u51fd\u6570<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/blog.iyatt.com\/?p=10802\/#%E9%9A%90%E5%87%BD%E6%95%B0\" >\u9690\u51fd\u6570<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/blog.iyatt.com\/?p=10802\/#%E5%8F%82%E6%95%B0%E6%96%B9%E7%A8%8B\" >\u53c2\u6570\u65b9\u7a0b<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h1><span class=\"ez-toc-section\" id=\"%E5%AE%9A%E4%B9%89\"><\/span>\u5b9a\u4e49<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u51fd\u6570\u7684\u5bfc\u6570\u4f9d\u7136\u53ef\u5bfc\uff0c\u90a3\u4e48\u51fd\u6570\u5bfc\u6570\u7684\u5bfc\u6570\u5c31\u662f<code>\u4e8c\u9636\u5bfc\u6570<\/code>\uff0c\u8bb0\u4f5c<code class=\"katex-inline\">y''\uff0cf''(x)\uff0c\\frac{d^2y}{dx^2}\uff0c\\frac{d^2f(x)}{dx^2}<\/code><br \/>\n\u4e8c\u9636\u5bfc\u6570\u7684\u5bfc\u6570\uff0c\u53eb<code>\u4e09\u9636\u5bfc\u6570<\/code>\uff0c\u8bb0\u4f5c<code class=\"katex-inline\">y'''\uff0cf'''(x)\uff0c\\frac{d^3y}{dx^3}\uff0c\\frac{d^3f(x)}{dx^3}<\/code><br \/>\n\u4e09\u9636\u5bfc\u6570\u7684\u5bfc\u6570\uff0c\u53eb<code>\u56db\u9636\u5bfc\u6570<\/code>\uff0c\u8bb0\u4f5c<code class=\"katex-inline\">y^{(4)}\uff0cf^{(4)}(x)\uff0c\\frac{d^4y}{dx^4}\uff0c\\frac{d^4f(x)}{dx^4}<\/code><br \/>\n&#8230;<br \/>\n<code>(n-1)<\/code>\u9636\u5bfc\u6570\u7684\u5bfc\u6570\uff0c\u53eb\u505a<code>n\u9636\u5bfc\u6570<\/code>\uff0c\u8bb0\u4f5c<code class=\"katex-inline\">y^{(n)}\uff0cf^{(n)}(x)\uff0c\\frac{d^ny}{dx^n}\uff0c\\frac{d^nf(x)}{dx^n}<\/code><\/p>\n<p>\u4e00\u9636\u6c42\u5bfc\u53c2\u8003\uff1a<a href=\"https:\/\/blog.iyatt.com\/?p=7986\">https:\/\/blog.iyatt.com\/?p=7986<\/a><\/p>\n<h1><span class=\"ez-toc-section\" id=\"%E6%B1%82%E5%AF%BC%E8%AE%A1%E7%AE%97\"><\/span>\u6c42\u5bfc\u8ba1\u7b97<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<h2><span class=\"ez-toc-section\" id=\"%E7%9B%B4%E6%8E%A5%E6%B3%95\"><\/span>\u76f4\u63a5\u6cd5<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u5148\u6c42\u51fa<code>y&#039;\uff0cy&#039;&#039;\uff0cy&#039;&#039;&#039;<\/code>\u7b49\uff0c\u627e\u51fa\u5bfc\u6570\u89c4\u5f8b\uff0c\u5199\u51fa<code class=\"katex-inline\">y^{(n)}<\/code>\u7ed3\u679c<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E9%97%B4%E6%8E%A5%E6%B3%95\"><\/span>\u95f4\u63a5\u6cd5<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u5229\u7528\u5df2\u77e5\u9ad8\u9636\u5bfc\u6570\u516c\u5f0f\u3001\u8fd0\u7b97\u6cd5\u5219\uff0c\u901a\u8fc7\u5c06\u51fd\u6570\u6052\u7b49\u53d8\u5f62\u3001\u53d8\u91cf\u66ff\u6362\u7b49\u6c42\u51fa\u9ad8\u9636\u5bfc\u6570\u7ed3\u679c\u3002<\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E5%9B%9B%E5%88%99%E8%BF%90%E7%AE%97%E6%B3%95%E5%88%99\"><\/span>\u56db\u5219\u8fd0\u7b97\u6cd5\u5219<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<pre><code class=\"language-katex\">\\begin{array}{l}\nu(x)\uff0cv(x)\u90fdn\u9636\u53ef\u5bfc \\\\\n[u(x)\\pm v(x)]^{(n)}=u^{(n)}(x)\\pm v^{n}(x) \\\\\n[C\\cdot u(x)]^{(n)}=C\\cdot u^{(n)}(x) \\\\\n[u(x)v(x)]^{(n)}=\\sum_{i=0}^nC_n^iu^{(n-i)}(x)\\cdot v^{(i)}(x)\uff0c\u5176\u4e2du^{(0)}(x)=u(x)\uff0cv^{(0)}(x)=v(x) \u2014\u2014\u83b1\u5e03\u5c3c\u8328\u516c\u5f0f\\\\\n\\end{array}<\/code><\/pre>\n<p>\u7ec4\u5408\u8ba1\u7b97\u516c\u5f0f\u53c2\u8003\uff1a<a href=\"https:\/\/blog.iyatt.com\/?p=10812\">https:\/\/blog.iyatt.com\/?p=10812<\/a><\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E5%B8%B8%E7%94%A8%E9%AB%98%E9%98%B6%E5%AF%BC%E6%95%B0%E5%85%AC%E5%BC%8F\"><\/span>\u5e38\u7528\u9ad8\u9636\u5bfc\u6570\u516c\u5f0f<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<pre><code class=\"language-katex\">\\begin{array}{l l}\n(x^n)^{(n)}=n! \\ \\ \\ (e^x)^{(n)}=e^x &amp; (a^x)^{(n)}=a^x\\ln^na \\\\\n(\\sin x)^{(n)}=\\sin(x+\\frac{\\pi}{2}n) &amp; (\\cos x)^{(n)}=\\cos(x+\\frac{\\pi}{2}n) \\\\\n(\\sin kx)^{(n)}=k^n\\sin(kx+\\frac{\\pi}{2}n) &amp; (\\cos kx)^{(n)}=k^n\\cos(kx+\\frac{\\pi}{2}n) \\\\\n(\\ln x)^{(n)}=(-1)^{(n-1)}\\frac{(n-1)!}{x^n} &amp; (\\frac{1}{ax+b})^{(n)}=(-1)^n\\frac{a^n\\cdot n!}{(ax+b)^{(n+1)}} \\\\\n\\end{array}<\/code><\/pre>\n<h1><span class=\"ez-toc-section\" id=\"%E9%83%A8%E5%88%86%E5%87%BD%E6%95%B0%E4%BA%8C%E9%98%B6%E5%AF%BC%E6%95%B0%E6%B1%82%E6%B3%95\"><\/span>\u90e8\u5206\u51fd\u6570\u4e8c\u9636\u5bfc\u6570\u6c42\u6cd5<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<h2><span class=\"ez-toc-section\" id=\"%E6%8A%BD%E8%B1%A1%E5%A4%8D%E5%90%88%E5%87%BD%E6%95%B0\"><\/span>\u62bd\u8c61\u590d\u5408\u51fd\u6570<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\begin{array}{l}\ny=f[\\phi(x)]\uff0c\u6c42\\frac{d^2y}{dx^2} \\\\\n\u5148\u6c42\u4e00\u9636\u5bfc\uff1a\\frac{dy}{dx}=f&#039;[\\phi (x)]\\phi&#039;(x) \\\\\n\u518d\u6c42\u4e8c\u9636\u5bfc\uff1a\\frac{d^2y}{dx^2}=f&#039;&#039;[\\phi(x)][\\phi&#039;(x)]^2+f&#039;[\\phi(x)]\\phi&#039;&#039;(x) \\\\\n\\\\\n\u4f8b \\\\\n[\\ln f(x)]&#039;=\\frac{f&#039;(x)}{f(x)} \\\\\n[\\ln f(x)]&#039;&#039;=\\frac{f&#039;&#039;(x)f(x)-[f&#039;(x)]^2}{f^2(x)}\n\\end{array}<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E9%9A%90%E5%87%BD%E6%95%B0\"><\/span>\u9690\u51fd\u6570<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u5148\u6c42\u4e00\u9636\u5bfc\uff0c\u518d\u4e8c\u9636\u5bfc\uff0c\u4e8c\u9636\u5bfc\u5f97\u5230\u7684\u5f0f\u5b50\u53ef\u80fd\u542b\u6709\u4e00\u9636\u5bfc\uff0c\u5c06\u4e00\u9636\u5bfc\u7ed3\u679c\u4ee3\u5165\uff0c\u6574\u7406\u5f97\u5230\u4e8c\u9636\u5bfc<\/p>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n\u4f8b \\\\\n\u7531\u65b9\u7a0b e^y+xy=e \u6c42 \\frac{d^2y}{dx^2} \\\\\n\\begin{aligned}\ne^yy&#039;+y+xy&#039;&amp;=0 \\\\\ny&#039;&amp;=-\\frac{y}{x+e^y}\n\\end{aligned}\n\\\\\n\\begin{aligned}\ny&#039;&#039;&amp;=-\\frac{y&#039;(x+e^y)-y(1+e^yy&#039;)}{(x+e^y)^2} \\\\\n\n&amp;=\\frac{2y-\\frac{y^2e^y}{x+e^y}}{(x+e^y)^2} \\\\\n&amp;=\\frac{2xy+2ye^y-y^22^2}{(x+e^y)^3}\n\\end{aligned}\n\\end{array}<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E5%8F%82%E6%95%B0%E6%96%B9%E7%A8%8B\"><\/span>\u53c2\u6570\u65b9\u7a0b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u7531\u53c2\u6570\u65b9\u7a0b<\/p>\n<pre><code class=\"language-katex\">\\left \\{\n\\begin{array}{l}\nx=\\phi(t) \\\\\ny=\\psi(t) \\\\\n\\end{array}\n\\right .<\/code><\/pre>\n<p>\u6240\u786e\u5b9a\u7684\u51fd\u6570<code>y=y(x)<\/code>\u7684\u4e8c\u9636\u5bfc\u6570<code class=\"katex-inline\">\\frac{d^2y}{dx^2}<\/code>\uff1a<\/p>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n\u5148\u6c42\u4e00\u9636\u5bfc\u6570\uff1a\\\\\n\\frac{dy}{dx}=\\frac{\\psi&#039;(t)}{\\phi&#039;(t)} \\\\\n\\\\\n\u518d\u6c42\u4e8c\u9636\u5bfc\u6570\uff1a\\\\\n\\begin{aligned}\n\\frac{d^2y}{dt^2}&amp;=\\frac{[\\frac{\\psi&#039;(t)}{\\phi&#039;(t)}]&#039;}{\\phi&#039;(t)} \\\\\n&amp;=\\frac{\\frac{\\psi&#039;&#039;(t)\\phi&#039;(t)-\\psi&#039;(t)\\phi&#039;&#039;(t)}{[\\phi&#039;(t)]^2}}{\\phi&#039;(t)} \\\\\n&amp;=\\frac{\\psi&#039;&#039;(t)\\phi&#039;(t)-\\psi&#039;(t)\\phi&#039;&#039;(t)}{[\\phi&#039;(t)]^3}\n\\end{aligned}\n\\end{array}<\/code><\/pre>\n<p>\u4f8b\uff082013\uff0c\u6570\u4e00\uff09<\/p>\n<pre><code class=\"language-katex\">\\left \\{\n\\begin{array}{l}\nx=\\sin t \\\\\ny=t\\sin t+\\cos t\n\\end{array}\n\\right .<\/code><\/pre>\n<p>\uff0c\u8ba1\u7b97<code class=\"katex-inline\">\\frac{d^2y}{dx^2}|_{t=\\frac{\\pi}{4}}<\/code>\u3002<\/p>\n<pre><code class=\"language-katex\">\\begin{array}{l}\ndy=t\\cos t\\ dt \\\\\ndx=\\cos t\\ dt \\\\\n\\\\\n\\frac{dy}{dx}=t \\\\\n\\\\\n\\frac{d^2y}{dx^2}=\\frac{[\\frac{dy}{dx}]&#039;}{dx}=\\frac{1}{\\cos t} \\\\\n\\\\\n\\frac{d^2y}{dx^2} |_{t=\\frac{\\pi}{4}}=\\sqrt2\n\\end{array}<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"<p>\u5b9a\u4e49 \u51fd\u6570\u7684\u5bfc\u6570\u4f9d\u7136\u53ef\u5bfc\uff0c\u90a3\u4e48\u51fd\u6570\u5bfc\u6570\u7684\u5bfc\u6570\u5c31\u662f\u4e8c\u9636\u5bfc\u6570\uff0c\u8bb0\u4f5cy&#8221;\uff0cf&#8221;(x)\uff0c\\frac{d^2y}{d [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"zakra_page_container_layout":"customizer","zakra_page_sidebar_layout":"customizer","zakra_remove_content_margin":false,"zakra_sidebar":"customizer","zakra_transparent_header":"customizer","zakra_logo":0,"zakra_main_header_style":"default","zakra_menu_item_color":"","zakra_menu_item_hover_color":"","zakra_menu_item_active_color":"","zakra_menu_active_style":"","zakra_page_header":true,"_lmt_disableupdate":"no","_lmt_disable":"no","footnotes":""},"categories":[1,612],"tags":[],"class_list":["post-10802","post","type-post","status-publish","format-standard","hentry","category-all","category-612"],"modified_by":"IYATT-yx","_links":{"self":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/10802","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=10802"}],"version-history":[{"count":0,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/10802\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10802"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10802"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10802"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}