{"id":685,"date":"2021-05-10T18:59:42","date_gmt":"2021-05-10T18:59:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=685"},"modified":"2021-11-17T02:50:23","modified_gmt":"2021-11-17T02:50:23","slug":"summary-of-first-order-linear-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-first-order-linear-equations\/","title":{"raw":"Summary of First-order Linear Equations","rendered":"Summary of First-order Linear Equations"},"content":{"raw":"<section id=\"fs-id1170572517580\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1170572517587\" data-bullet-style=\"bullet\">\r\n \t<li>Any first-order linear differential equation can be written in the form [latex]y^{\\prime} +p\\left(x\\right)y=q\\left(x\\right)[\/latex].<\/li>\r\n \t<li>We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value.<\/li>\r\n \t<li>Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1170572411085\" class=\"key-equations\" data-depth=\"1\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170572411092\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">standard form<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]y^{\\prime} +p\\left(x\\right)y=q\\left(x\\right)[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">integrating factor<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\mu \\left(x\\right)={e}^{\\displaystyle\\int p\\left(x\\right)dx}[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section id=\"fs-id1170572604389\" class=\"section-exercises\" data-depth=\"1\"><\/section><section id=\"fs-id1170572115950\" class=\"review-exercises\" data-depth=\"1\">\r\n<div id=\"fs-id1170571832364\" data-type=\"exercise\"><\/div>\r\n<\/section>\r\n<div data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170571813852\">\r\n \t<dt>integrating factor<\/dt>\r\n \t<dd id=\"fs-id1170571813858\">any function [latex]f\\left(x\\right)[\/latex] that is multiplied on both sides of a differential equation to make the side involving the unknown function equal to the derivative of a product of two functions<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571813876\">\r\n \t<dt>linear<\/dt>\r\n \t<dd id=\"fs-id1170571813882\">description of a first-order differential equation that can be written in the form [latex]a\\left(x\\right){y}^{\\prime }+b\\left(x\\right)y=c\\left(x\\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572386140\">\r\n \t<dt>standard form<\/dt>\r\n \t<dd id=\"fs-id1170572386145\">the form of a first-order linear differential equation obtained by writing the differential equation in the form [latex]y^{\\prime} +p\\left(x\\right)y=q\\left(x\\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1170572517580\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1170572517587\" data-bullet-style=\"bullet\">\n<li>Any first-order linear differential equation can be written in the form [latex]y^{\\prime} +p\\left(x\\right)y=q\\left(x\\right)[\/latex].<\/li>\n<li>We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value.<\/li>\n<li>Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1170572411085\" class=\"key-equations\" data-depth=\"1\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170572411092\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">standard form<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]y^{\\prime} +p\\left(x\\right)y=q\\left(x\\right)[\/latex]<\/li>\n<li><strong data-effect=\"bold\">integrating factor<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\mu \\left(x\\right)={e}^{\\displaystyle\\int p\\left(x\\right)dx}[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-id1170572604389\" class=\"section-exercises\" data-depth=\"1\"><\/section>\n<section id=\"fs-id1170572115950\" class=\"review-exercises\" data-depth=\"1\">\n<div id=\"fs-id1170571832364\" data-type=\"exercise\"><\/div>\n<\/section>\n<div data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170571813852\">\n<dt>integrating factor<\/dt>\n<dd id=\"fs-id1170571813858\">any function [latex]f\\left(x\\right)[\/latex] that is multiplied on both sides of a differential equation to make the side involving the unknown function equal to the derivative of a product of two functions<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571813876\">\n<dt>linear<\/dt>\n<dd id=\"fs-id1170571813882\">description of a first-order differential equation that can be written in the form [latex]a\\left(x\\right){y}^{\\prime }+b\\left(x\\right)y=c\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572386140\">\n<dt>standard form<\/dt>\n<dd id=\"fs-id1170572386145\">the form of a first-order linear differential equation obtained by writing the differential equation in the form [latex]y^{\\prime} +p\\left(x\\right)y=q\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n<\/div>\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-685\">\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 2. <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-2\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-2\/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-2\/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":416434,"menu_order":20,"template":"","meta":{"_candela_citation":"{\"1\":{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) 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