{"id":683,"date":"2021-05-10T18:59:10","date_gmt":"2021-05-10T18:59:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=683"},"modified":"2021-11-17T02:47:12","modified_gmt":"2021-11-17T02:47:12","slug":"summary-of-separable-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-separable-equations\/","title":{"raw":"Summary of Separable Equations","rendered":"Summary of Separable Equations"},"content":{"raw":"<section id=\"fs-id1170573604128\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1170573604134\" data-bullet-style=\"bullet\">\r\n \t<li>A separable differential equation is any equation that can be written in the form [latex]y^{\\prime} =f\\left(x\\right)g\\left(y\\right)[\/latex].<\/li>\r\n \t<li>The method of separation of variables is used to find the general solution to a separable differential equation.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1170571346190\" class=\"key-equations\" data-depth=\"1\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170571346197\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Separable differential equation<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]{y}^{\\prime }=f\\left(x\\right)g\\left(y\\right)[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Solution concentration<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\frac{du}{dt}=\\text{INFLOW RATE}-\\text{OUTFLOW RATE}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Newton\u2019s law of cooling<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\frac{dT}{dt}=k\\left(T-{T}_{s}\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section id=\"fs-id1170571116657\" class=\"section-exercises\" data-depth=\"1\">\r\n<div data-type=\"exercise\"><\/div>\r\n<div id=\"fs-id1170573742107\" data-type=\"exercise\"><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Glossary<\/span><\/div>\r\n<\/section>\r\n<div data-type=\"glossary\">\r\n<dl id=\"fs-id1170571243486\">\r\n \t<dt>autonomous differential equation<\/dt>\r\n \t<dd id=\"fs-id1170571243492\">an equation in which the right-hand side is a function of [latex]y[\/latex] alone<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170571243500\">\r\n \t<dt>separable differential equation<\/dt>\r\n \t<dd id=\"fs-id1170571243506\">any equation that can be written in the form [latex]y^{\\prime} =f\\left(x\\right)g\\left(y\\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573742138\">\r\n \t<dt>separation of variables<\/dt>\r\n \t<dd id=\"fs-id1170573742143\">a method used to solve a separable differential equation<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1170573604128\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1170573604134\" data-bullet-style=\"bullet\">\n<li>A separable differential equation is any equation that can be written in the form [latex]y^{\\prime} =f\\left(x\\right)g\\left(y\\right)[\/latex].<\/li>\n<li>The method of separation of variables is used to find the general solution to a separable differential equation.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1170571346190\" class=\"key-equations\" data-depth=\"1\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170571346197\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Separable differential equation<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]{y}^{\\prime }=f\\left(x\\right)g\\left(y\\right)[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Solution concentration<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\frac{du}{dt}=\\text{INFLOW RATE}-\\text{OUTFLOW RATE}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Newton\u2019s law of cooling<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\frac{dT}{dt}=k\\left(T-{T}_{s}\\right)[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-id1170571116657\" class=\"section-exercises\" data-depth=\"1\">\n<div data-type=\"exercise\"><\/div>\n<div id=\"fs-id1170573742107\" data-type=\"exercise\"><span style=\"color: #077fab; font-size: 1.15em; font-weight: 600;\">Glossary<\/span><\/div>\n<\/section>\n<div data-type=\"glossary\">\n<dl id=\"fs-id1170571243486\">\n<dt>autonomous differential equation<\/dt>\n<dd id=\"fs-id1170571243492\">an equation in which the right-hand side is a function of [latex]y[\/latex] alone<\/dd>\n<\/dl>\n<dl id=\"fs-id1170571243500\">\n<dt>separable differential equation<\/dt>\n<dd id=\"fs-id1170571243506\">any equation that can be written in the form [latex]y^{\\prime} =f\\left(x\\right)g\\left(y\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170573742138\">\n<dt>separation of variables<\/dt>\n<dd id=\"fs-id1170573742143\">a method used to solve a separable differential equation<\/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-683\">\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":12,"template":"","meta":{"_candela_citation":"{\"1\":{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/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-683","chapter","type-chapter","status-publish","hentry"],"part":159,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/683","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/416434"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/683\/revisions"}],"predecessor-version":[{"id":2557,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/683\/revisions\/2557"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/159"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/683\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=683"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=683"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=683"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=683"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}