{"id":86,"date":"2021-07-30T17:11:15","date_gmt":"2021-07-30T17:11:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=86"},"modified":"2022-11-01T22:47:48","modified_gmt":"2022-11-01T22:47:48","slug":"summary-of-nonhomogeneous-linear-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-nonhomogeneous-linear-equations\/","title":{"raw":"Summary of Nonhomogeneous Linear Equations","rendered":"Summary of Nonhomogeneous Linear Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<div class=\"PageContent-ny9bj0-0 gLsPXs\" tabindex=\"0\">\r\n<div id=\"main-content\" class=\"MainContent__HideOutline-sc-6yy1if-0 bdVAq sc-bdVaJa ibTDPh\" tabindex=\"-1\" data-dynamic-style=\"false\">\r\n<div id=\"composite-page-27\" class=\"os-eoc os-key-concepts-container\" data-type=\"composite-page\" data-uuid-key=\".key-concepts\">\r\n<div class=\"os-key-concepts\">\r\n<div class=\"os-section-area\"><section id=\"fs-id1170571115987\" class=\"key-concepts\" data-depth=\"1\">\r\n<ul id=\"fs-id1170571115993\" data-bullet-style=\"bullet\">\r\n \t<li>To solve a nonhomogeneous linear second-order differential equation, first find the general solution to the complementary equation, then find a particular solution to the nonhomogeneous equation.<\/li>\r\n \t<li>Let [latex]y_{p}(x)[\/latex] be any particular solution to the nonhomogeneous linear differential equation\u00a0[latex]a_{2}(x)y^{\\prime\\prime}+a_{1}(x)y^{\\prime}+a_{0}(x)y=r(x)[\/latex], and let\u00a0[latex]c_{1}y_{1}(x)+c_{2}y_{2}(x)[\/latex]\u00a0denote the general solution to the complementary equation. Then, the general solution to the nonhomogeneous equation is given by [latex]y(x)=c_{1}y_{1}(x)+c_{2}y_{2}(x)+y_{p}(x)[\/latex].<\/li>\r\n \t<li>When [latex]r(x)[\/latex] is a combination of polynomials, exponential functions, sines, and cosines, use the method of undetermined coefficients to find the particular solution. To use this method, assume a solution in the same form as\u00a0[latex]r(x)[\/latex],\u00a0multiplying by\u00a0[latex]x[\/latex]\u00a0as necessary until the assumed solution is linearly independent of the general solution to the complementary equation. Then, substitute the assumed solution into the differential equation to find values for the coefficients.<\/li>\r\n \t<li>When [latex]r(x)[\/latex]\u00a0is\u00a0<em data-effect=\"italics\">not<\/em>\u00a0a combination of polynomials, exponential functions, or sines and cosines, use the method of variation of parameters to find the particular solution. This method involves using Cramer\u2019s rule or another suitable technique to find functions [latex]u^{\\prime}(x)[\/latex] and\u00a0[latex]v^{\\prime}(x)[\/latex] satisfying [latex]\\begin{array}{c} \\hfill u^{\\prime}y_{1}+v^{\\prime}y_{2} &amp;= 0 \\hfill \\\\u^{\\prime}y_{1}^{\\prime}+v^{\\prime}y_{2}^{\\prime} &amp;= r(x) \\hfill\\end{array}[\/latex]. Then,\u00a0[latex]y_{p}(x)=u(x)y_{1}(x)+v(x)y_{2}(x)[\/latex] is a particular solution to the differential equation.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Complementary equation\r\n<\/strong>[latex]a_{2}(x)y^{\\prime\\prime}+a_{1}(x)y^{\\prime}+a_{0}(x)y=0[\/latex]<\/li>\r\n \t<li><strong>General solution to a nonhomogeneous linear differential equation<\/strong>\r\n[latex]y(x)=c_{1}y_{1}(x)+c_{2}y_{2}(x)+y_{p}(x)[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>complementary equation<\/dt>\r\n \t<dd>for the nonhomogeneous linear differential equation [latex]a_{2}(x)y^{\\prime\\prime}+a_{1}(x)y^{\\prime}+a_{0}(x)y=r(x)[\/latex]\u00a0the associated homogeneous equation, called the <em data-effect=\"italics\">complementary equation<\/em>, is\u00a0[latex]a_{2}(x)y^{\\prime\\prime}+a_{1}(x)y^{\\prime}+a_{0}(x)y=0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>method of undetermined coefficients<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a method that involves making a guess about the form of the particular solution, then solving for the coefficients in the guess<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>method of variation of parameters<\/dt>\r\n \t<dd>a method that involves looking for particular solutions in the form [latex]y_{p}(x)=u(x)y_{1}(x)+v(x)y_{2}(x)[\/latex], where [latex]y_{1}[\/latex]\u00a0and [latex]y_{2}[\/latex]\u00a0are linearly independent solutions to the complementary equations, and then solving a system of equations to find [latex]u(x)[\/latex]\u00a0and\u00a0[latex]v(x)[\/latex].<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>particular solution<\/dt>\r\n \t<dd>a solution [latex]y_{p}(x)[\/latex]\u00a0of a differential equation that contains no arbitrary constants<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<div class=\"PageContent-ny9bj0-0 gLsPXs\" tabindex=\"0\">\n<div id=\"main-content\" class=\"MainContent__HideOutline-sc-6yy1if-0 bdVAq sc-bdVaJa ibTDPh\" tabindex=\"-1\" data-dynamic-style=\"false\">\n<div id=\"composite-page-27\" class=\"os-eoc os-key-concepts-container\" data-type=\"composite-page\" data-uuid-key=\".key-concepts\">\n<div class=\"os-key-concepts\">\n<div class=\"os-section-area\">\n<section id=\"fs-id1170571115987\" class=\"key-concepts\" data-depth=\"1\">\n<ul id=\"fs-id1170571115993\" data-bullet-style=\"bullet\">\n<li>To solve a nonhomogeneous linear second-order differential equation, first find the general solution to the complementary equation, then find a particular solution to the nonhomogeneous equation.<\/li>\n<li>Let [latex]y_{p}(x)[\/latex] be any particular solution to the nonhomogeneous linear differential equation\u00a0[latex]a_{2}(x)y^{\\prime\\prime}+a_{1}(x)y^{\\prime}+a_{0}(x)y=r(x)[\/latex], and let\u00a0[latex]c_{1}y_{1}(x)+c_{2}y_{2}(x)[\/latex]\u00a0denote the general solution to the complementary equation. Then, the general solution to the nonhomogeneous equation is given by [latex]y(x)=c_{1}y_{1}(x)+c_{2}y_{2}(x)+y_{p}(x)[\/latex].<\/li>\n<li>When [latex]r(x)[\/latex] is a combination of polynomials, exponential functions, sines, and cosines, use the method of undetermined coefficients to find the particular solution. To use this method, assume a solution in the same form as\u00a0[latex]r(x)[\/latex],\u00a0multiplying by\u00a0[latex]x[\/latex]\u00a0as necessary until the assumed solution is linearly independent of the general solution to the complementary equation. Then, substitute the assumed solution into the differential equation to find values for the coefficients.<\/li>\n<li>When [latex]r(x)[\/latex]\u00a0is\u00a0<em data-effect=\"italics\">not<\/em>\u00a0a combination of polynomials, exponential functions, or sines and cosines, use the method of variation of parameters to find the particular solution. This method involves using Cramer\u2019s rule or another suitable technique to find functions [latex]u^{\\prime}(x)[\/latex] and\u00a0[latex]v^{\\prime}(x)[\/latex] satisfying [latex]\\begin{array}{c} \\hfill u^{\\prime}y_{1}+v^{\\prime}y_{2} &= 0 \\hfill \\\\u^{\\prime}y_{1}^{\\prime}+v^{\\prime}y_{2}^{\\prime} &= r(x) \\hfill\\end{array}[\/latex]. Then,\u00a0[latex]y_{p}(x)=u(x)y_{1}(x)+v(x)y_{2}(x)[\/latex] is a particular solution to the differential equation.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Complementary equation<br \/>\n<\/strong>[latex]a_{2}(x)y^{\\prime\\prime}+a_{1}(x)y^{\\prime}+a_{0}(x)y=0[\/latex]<\/li>\n<li><strong>General solution to a nonhomogeneous linear differential equation<\/strong><br \/>\n[latex]y(x)=c_{1}y_{1}(x)+c_{2}y_{2}(x)+y_{p}(x)[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>complementary equation<\/dt>\n<dd>for the nonhomogeneous linear differential equation [latex]a_{2}(x)y^{\\prime\\prime}+a_{1}(x)y^{\\prime}+a_{0}(x)y=r(x)[\/latex]\u00a0the associated homogeneous equation, called the <em data-effect=\"italics\">complementary equation<\/em>, is\u00a0[latex]a_{2}(x)y^{\\prime\\prime}+a_{1}(x)y^{\\prime}+a_{0}(x)y=0[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>method of undetermined coefficients<\/dt>\n<dd><span style=\"font-size: 1em;\">a method that involves making a guess about the form of the particular solution, then solving for the coefficients in the guess<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>method of variation of parameters<\/dt>\n<dd>a method that involves looking for particular solutions in the form [latex]y_{p}(x)=u(x)y_{1}(x)+v(x)y_{2}(x)[\/latex], where [latex]y_{1}[\/latex]\u00a0and [latex]y_{2}[\/latex]\u00a0are linearly independent solutions to the complementary equations, and then solving a system of equations to find [latex]u(x)[\/latex]\u00a0and\u00a0[latex]v(x)[\/latex].<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>particular solution<\/dt>\n<dd>a solution [latex]y_{p}(x)[\/latex]\u00a0of a differential equation that contains no arbitrary constants<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-86\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-86","chapter","type-chapter","status-publish","hentry"],"part":25,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/86","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/86\/revisions"}],"predecessor-version":[{"id":3798,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/86\/revisions\/3798"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/25"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/86\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=86"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=86"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=86"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=86"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}