{"id":832,"date":"2025-10-09T15:42:11","date_gmt":"2025-10-09T15:42:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=832"},"modified":"2025-11-13T21:07:23","modified_gmt":"2025-11-13T21:07:23","slug":"series-solutions-of-differential-equations-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/series-solutions-of-differential-equations-fresh-take\/","title":{"raw":"Series Solutions of Differential Equations: Fresh Take","rendered":"Series Solutions of Differential Equations: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Solve differential equations using power series<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"problem-solving ui-has-child-title\" data-type=\"note\">Series Solutions of Differential Equations<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Some differential equations can be solved by assuming the solution has the form of a power series: [latex]y(x) = \\sum_{n=0}^{\\infty} a_nx^n[\/latex]. We find the coefficients [latex]a_n[\/latex] by substituting the series into the differential equation and matching coefficients of like powers of [latex]x[\/latex].<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>The Method<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Start by assuming [latex]y(x) = \\sum_{n=0}^{\\infty} a_nx^n[\/latex]. Differentiate term by term to get [latex]y'(x) = \\sum_{n=1}^{\\infty} na_nx^{n-1}[\/latex] and [latex]y''(x) = \\sum_{n=2}^{\\infty} n(n-1)a_nx^{n-2}[\/latex]. Substitute these expressions into the differential equation.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Re-indexing and Combining<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">After substitution, you'll have multiple summations with different starting indices and powers of [latex]x[\/latex]. Re-index the sums so they all have the same power of [latex]x[\/latex] and can be combined into a single summation. This typically involves replacing [latex]n[\/latex] with [latex]n+k[\/latex] for some integer [latex]k[\/latex] and adjusting the summation limits accordingly.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Finding the Coefficients<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Once combined into a single sum, the equation can only be satisfied if the coefficient of each power of [latex]x[\/latex] equals zero. This gives you a <strong>recurrence relation<\/strong>\u2014an equation relating [latex]a_n[\/latex] to earlier coefficients. Solve this relation to express all coefficients in terms of [latex]a_0[\/latex], [latex]a_1[\/latex], or other initial values.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Special Applications<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Power series solutions appear in many physical applications. The <strong>Bessel equation<\/strong> [latex]x^2y'' + xy' + (x^2 - n^2)y = 0[\/latex] arises in problems with cylindrical symmetry, such as vibrating drum heads and heat flow in cylinders. These equations often produce solutions involving even or odd powers only, simplifying the pattern of coefficients.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1170571291319\">Find a power series solution for the following differential equations.<\/p>\r\n\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]y^{\\prime}+2xy=0[\/latex]<\/li>\r\n \t<li>[latex](x+1)y^\\prime=3y[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"652781369\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"652781369\"]\r\n\r\na. [latex]y(x)=a_0\\displaystyle\\sum_{n=0}^\\infty\\frac{(-1)^n}{n!}x^{2n}=a_0e^{-x^2}[\/latex]\r\n\r\nb. [latex]y(x)=a_0(x+1)^3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8250347&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=d5VG6PM48vQ&amp;video_target=tpm-plugin-7w3p47zd-d5VG6PM48vQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP7.22a_transcript.html\">transcript for \u201cCP 7.22a\u201d here (opens in new window).<\/a><\/center><center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8250348&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=2ukZj1dveVs&amp;video_target=tpm-plugin-lphrhkde-2ukZj1dveVs\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP7.22b_transcript.html\">transcript for \u201cCP 7.22b\u201d here (opens in new window).<\/a><\/center><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Solve differential equations using power series<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"problem-solving ui-has-child-title\" data-type=\"note\">Series Solutions of Differential Equations<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Some differential equations can be solved by assuming the solution has the form of a power series: [latex]y(x) = \\sum_{n=0}^{\\infty} a_nx^n[\/latex]. We find the coefficients [latex]a_n[\/latex] by substituting the series into the differential equation and matching coefficients of like powers of [latex]x[\/latex].<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>The Method<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Start by assuming [latex]y(x) = \\sum_{n=0}^{\\infty} a_nx^n[\/latex]. Differentiate term by term to get [latex]y'(x) = \\sum_{n=1}^{\\infty} na_nx^{n-1}[\/latex] and [latex]y''(x) = \\sum_{n=2}^{\\infty} n(n-1)a_nx^{n-2}[\/latex]. Substitute these expressions into the differential equation.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Re-indexing and Combining<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">After substitution, you&#8217;ll have multiple summations with different starting indices and powers of [latex]x[\/latex]. Re-index the sums so they all have the same power of [latex]x[\/latex] and can be combined into a single summation. This typically involves replacing [latex]n[\/latex] with [latex]n+k[\/latex] for some integer [latex]k[\/latex] and adjusting the summation limits accordingly.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Finding the Coefficients<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Once combined into a single sum, the equation can only be satisfied if the coefficient of each power of [latex]x[\/latex] equals zero. This gives you a <strong>recurrence relation<\/strong>\u2014an equation relating [latex]a_n[\/latex] to earlier coefficients. Solve this relation to express all coefficients in terms of [latex]a_0[\/latex], [latex]a_1[\/latex], or other initial values.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Special Applications<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Power series solutions appear in many physical applications. The <strong>Bessel equation<\/strong> [latex]x^2y'' + xy' + (x^2 - n^2)y = 0[\/latex] arises in problems with cylindrical symmetry, such as vibrating drum heads and heat flow in cylinders. These equations often produce solutions involving even or odd powers only, simplifying the pattern of coefficients.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1170571291319\">Find a power series solution for the following differential equations.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]y^{\\prime}+2xy=0[\/latex]<\/li>\n<li>[latex](x+1)y^\\prime=3y[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q652781369\">Show Solution<\/button><\/p>\n<div id=\"q652781369\" class=\"hidden-answer\" style=\"display: none\">\n<p>a. [latex]y(x)=a_0\\displaystyle\\sum_{n=0}^\\infty\\frac{(-1)^n}{n!}x^{2n}=a_0e^{-x^2}[\/latex]<\/p>\n<p>b. [latex]y(x)=a_0(x+1)^3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8250347&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=d5VG6PM48vQ&amp;video_target=tpm-plugin-7w3p47zd-d5VG6PM48vQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP7.22a_transcript.html\">transcript for \u201cCP 7.22a\u201d here (opens in new window).<\/a><\/div>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8250348&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=2ukZj1dveVs&amp;video_target=tpm-plugin-lphrhkde-2ukZj1dveVs\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP7.22b_transcript.html\">transcript for \u201cCP 7.22b\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":25,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-832","chapter","type-chapter","status-publish","hentry"],"part":440,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/832","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/832\/revisions"}],"predecessor-version":[{"id":2271,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/832\/revisions\/2271"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/440"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/832\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=832"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=832"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=832"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=832"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}