{"id":84,"date":"2021-07-30T17:11:00","date_gmt":"2021-07-30T17:11:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=84"},"modified":"2022-11-01T22:40:42","modified_gmt":"2022-11-01T22:40:42","slug":"summary-of-second-order-linear-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-second-order-linear-equations\/","title":{"raw":"Summary of Second-Order Linear Equations","rendered":"Summary of Second-Order Linear Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>Second-order differential equations can be classified as linear or nonlinear, homogeneous or nonhomogeneous.<\/li>\r\n \t<li>To find a general solution for a homogeneous second-order differential equation, we must find two linearly independent solutions. If\u00a0[latex]y_{1}(x)[\/latex] and\u00a0[latex]y_{2}(x)[\/latex]\u00a0are linearly independent solutions to a second-order, linear, homogeneous differential equation, then the general solution is given by\u00a0[latex]y(x)=c_{1}y_{1}(x)+c_{2}y_{2}(x)[\/latex].<\/li>\r\n \t<li>To solve homogeneous second-order differential equations with constant coefficients, find the roots of the characteristic equation. The form of the general solution varies depending on whether the characteristic equation has distinct, real roots; a single, repeated real root; or complex conjugate roots.<\/li>\r\n \t<li>Initial conditions or boundary conditions can then be used to find the specific solution to a differential equation that satisfies those conditions, except when there is no solution or infinitely many solutions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Linear second-order differential equation\r\n<\/strong>[latex]a_{2}(x)y^{\\prime\\prime}+a_{1}(x)y^{\\prime}+a_{0}(x)y=r(x)[\/latex]<\/li>\r\n \t<li><strong>Second-order equation with constant coefficients<\/strong>\r\n[latex]ay^{\\prime\\prime}+by^{\\prime}+cy=0[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>boundary conditions<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">the conditions that give the state of a system at different times, such as the position of a spring-mass system at two different times<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>boundary-value problem<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a differential equation with associated boundary conditions<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>characteristic equation<\/dt>\r\n \t<dd>the equation [latex]a\\lambda^{2}+b\\lambda+c=0[\/latex]<strong>\u00a0<\/strong>for the differential equation [latex]ay^{\\prime\\prime}+by^{\\prime}+cy=0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>homogeneous linear equation<\/dt>\r\n \t<dd>a second-order differential equation that can be written in the form [latex]a_{2}(x)y^{\\prime\\prime}+a_{1}(x)y^{\\prime}+a_{0}(x)y=r(x)[\/latex]\u00a0but [latex]r(x)=0[\/latex]\u00a0for every value of [latex]x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>linearly dependent<\/dt>\r\n \t<dd>a set of function [latex]f_{1}(x),f_{2}(x),\\ldots f_{n}(x)[\/latex]\u00a0for which there are constants [latex]c_{1},c_{2},\\ldots c_{n}[\/latex], not all zero, such that [latex]c_{1}f_{1}(x) + c_{2}f_{2}(x) + {\\cdots}+c_{n}f_{n}(x) = 0[\/latex] for all [latex]x[\/latex] in the interval of interest<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>linearly independent<\/dt>\r\n \t<dd>a set of function [latex]f_{1}(x),f_{2}(x),\\ldots f_{n}(x)[\/latex]\u00a0for which there are no constants, such that\u00a0<span style=\"font-size: 1em;\">[latex]c_{1},c_{2},\\ldots c_{n}[\/latex], such that [latex]c_{1}f_{1}(x) + c_{2}f_{2}(x) + {\\cdots}+c_{n}f_{n}(x) = 0[\/latex]\u00a0<\/span><span style=\"font-size: 1em;\">for all [latex]x[\/latex]<\/span>\u00a0in the interval of interest<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>nonhomogeneous linear equation<\/dt>\r\n \t<dd>a second-order differential equation that can be written in the form [latex]a_{2}(x)y^{\\prime\\prime}+a_{1}(x)y^{\\prime}+a_{0}(x)y=r(x)[\/latex]\u00a0but [latex]r(x)\\ne 0[\/latex]\u00a0for some value of [latex]x[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>Second-order differential equations can be classified as linear or nonlinear, homogeneous or nonhomogeneous.<\/li>\n<li>To find a general solution for a homogeneous second-order differential equation, we must find two linearly independent solutions. If\u00a0[latex]y_{1}(x)[\/latex] and\u00a0[latex]y_{2}(x)[\/latex]\u00a0are linearly independent solutions to a second-order, linear, homogeneous differential equation, then the general solution is given by\u00a0[latex]y(x)=c_{1}y_{1}(x)+c_{2}y_{2}(x)[\/latex].<\/li>\n<li>To solve homogeneous second-order differential equations with constant coefficients, find the roots of the characteristic equation. The form of the general solution varies depending on whether the characteristic equation has distinct, real roots; a single, repeated real root; or complex conjugate roots.<\/li>\n<li>Initial conditions or boundary conditions can then be used to find the specific solution to a differential equation that satisfies those conditions, except when there is no solution or infinitely many solutions.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Linear second-order differential equation<br \/>\n<\/strong>[latex]a_{2}(x)y^{\\prime\\prime}+a_{1}(x)y^{\\prime}+a_{0}(x)y=r(x)[\/latex]<\/li>\n<li><strong>Second-order equation with constant coefficients<\/strong><br \/>\n[latex]ay^{\\prime\\prime}+by^{\\prime}+cy=0[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>boundary conditions<\/dt>\n<dd><span style=\"font-size: 1em;\">the conditions that give the state of a system at different times, such as the position of a spring-mass system at two different times<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>boundary-value problem<\/dt>\n<dd><span style=\"font-size: 1em;\">a differential equation with associated boundary conditions<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>characteristic equation<\/dt>\n<dd>the equation [latex]a\\lambda^{2}+b\\lambda+c=0[\/latex]<strong>\u00a0<\/strong>for the differential equation [latex]ay^{\\prime\\prime}+by^{\\prime}+cy=0[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>homogeneous linear equation<\/dt>\n<dd>a second-order differential equation that can be written in the form [latex]a_{2}(x)y^{\\prime\\prime}+a_{1}(x)y^{\\prime}+a_{0}(x)y=r(x)[\/latex]\u00a0but [latex]r(x)=0[\/latex]\u00a0for every value of [latex]x[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>linearly dependent<\/dt>\n<dd>a set of function [latex]f_{1}(x),f_{2}(x),\\ldots f_{n}(x)[\/latex]\u00a0for which there are constants [latex]c_{1},c_{2},\\ldots c_{n}[\/latex], not all zero, such that [latex]c_{1}f_{1}(x) + c_{2}f_{2}(x) + {\\cdots}+c_{n}f_{n}(x) = 0[\/latex] for all [latex]x[\/latex] in the interval of interest<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>linearly independent<\/dt>\n<dd>a set of function [latex]f_{1}(x),f_{2}(x),\\ldots f_{n}(x)[\/latex]\u00a0for which there are no constants, such that\u00a0<span style=\"font-size: 1em;\">[latex]c_{1},c_{2},\\ldots c_{n}[\/latex], such that [latex]c_{1}f_{1}(x) + c_{2}f_{2}(x) + {\\cdots}+c_{n}f_{n}(x) = 0[\/latex]\u00a0<\/span><span style=\"font-size: 1em;\">for all [latex]x[\/latex]<\/span>\u00a0in the interval of interest<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>nonhomogeneous linear equation<\/dt>\n<dd>a second-order differential equation that can be written in the form [latex]a_{2}(x)y^{\\prime\\prime}+a_{1}(x)y^{\\prime}+a_{0}(x)y=r(x)[\/latex]\u00a0but [latex]r(x)\\ne 0[\/latex]\u00a0for some value of [latex]x[\/latex]<\/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-84\">\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":6,"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-84","chapter","type-chapter","status-publish","hentry"],"part":25,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/84","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":15,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/84\/revisions"}],"predecessor-version":[{"id":3797,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/84\/revisions\/3797"}],"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\/84\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=84"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=84"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=84"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=84"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}