{"id":23509,"date":"2022-05-28T22:38:30","date_gmt":"2022-05-28T22:38:30","guid":{"rendered":"https:\/\/onlinemathtutor.co\/?page_id=23509"},"modified":"2023-04-21T00:47:51","modified_gmt":"2023-04-21T00:47:51","slug":"virtual-lessons-de-separable","status":"publish","type":"page","link":"https:\/\/onlinemathtutor.co\/course-content\/differential-equations-resources\/first-order-de\/virtual-lessons-de-separable\/","title":{"rendered":"Virtual Lessons &#8211; DE &#8211; [Separable]"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"23509\" class=\"elementor elementor-23509\" data-elementor-post-type=\"page\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4d071591 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4d071591\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1cb1f534\" data-id=\"1cb1f534\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7862cfac elementor-widget elementor-widget-heading\" data-id=\"7862cfac\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">Virtual Lessons: Separable DEs<\/h1>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6bb34d0c elementor-section-stretched elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6bb34d0c\" data-element_type=\"section\" data-settings=\"{&quot;stretch_section&quot;:&quot;section-stretched&quot;}\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1baed2c8\" data-id=\"1baed2c8\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-680e7b5 elementor-nav-menu__align-center elementor-nav-menu--stretch elementor-nav-menu__text-align-center animated-fast elementor-nav-menu--dropdown-tablet elementor-nav-menu--toggle elementor-nav-menu--burger elementor-widget elementor-widget-nav-menu\" data-id=\"680e7b5\" data-element_type=\"widget\" data-settings=\"{&quot;submenu_icon&quot;:{&quot;value&quot;:&quot;&lt;i class=\\&quot;fas fa-pencil-alt\\&quot;&gt;&lt;\\\/i&gt;&quot;,&quot;library&quot;:&quot;fa-solid&quot;},&quot;full_width&quot;:&quot;stretch&quot;,&quot;_animation&quot;:&quot;none&quot;,&quot;layout&quot;:&quot;horizontal&quot;,&quot;toggle&quot;:&quot;burger&quot;}\" data-widget_type=\"nav-menu.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t<nav aria-label=\"Menu\" class=\"elementor-nav-menu--main elementor-nav-menu__container elementor-nav-menu--layout-horizontal e--pointer-double-line e--animation-fast\">\n\t\t\t\t<ul id=\"menu-1-680e7b5\" class=\"elementor-nav-menu\"><li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-23492\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/differential-equations-resources\/first-order-de\/practice-problems-de-separable\/\" class=\"elementor-item\">Separable DEs<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-23427\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/differential-equations-resources\/first-order-de\/practice-problems-de-intfactor\/\" class=\"elementor-item\">Integrating Factor<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-23430\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/differential-equations-resources\/first-order-de\/practice-problems-de-exact\/\" class=\"elementor-item\">Exact DEs<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-23428\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/differential-equations-resources\/first-order-de\/practice-problems-de-firstordersub\/\" class=\"elementor-item\">Substitution<\/a><\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t\t<div class=\"elementor-menu-toggle\" role=\"button\" tabindex=\"0\" aria-label=\"Menu Toggle\" aria-expanded=\"false\">\n\t\t\t<i aria-hidden=\"true\" role=\"presentation\" class=\"elementor-menu-toggle__icon--open eicon-menu-bar\"><\/i><i aria-hidden=\"true\" role=\"presentation\" class=\"elementor-menu-toggle__icon--close eicon-close\"><\/i>\t\t<\/div>\n\t\t\t\t\t<nav class=\"elementor-nav-menu--dropdown elementor-nav-menu__container\" aria-hidden=\"true\">\n\t\t\t\t<ul id=\"menu-2-680e7b5\" class=\"elementor-nav-menu\"><li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-23492\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/differential-equations-resources\/first-order-de\/practice-problems-de-separable\/\" class=\"elementor-item\" tabindex=\"-1\">Separable DEs<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-23427\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/differential-equations-resources\/first-order-de\/practice-problems-de-intfactor\/\" class=\"elementor-item\" tabindex=\"-1\">Integrating Factor<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-23430\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/differential-equations-resources\/first-order-de\/practice-problems-de-exact\/\" class=\"elementor-item\" tabindex=\"-1\">Exact DEs<\/a><\/li>\n<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-23428\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/differential-equations-resources\/first-order-de\/practice-problems-de-firstordersub\/\" class=\"elementor-item\" tabindex=\"-1\">Substitution<\/a><\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-61878a72 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"61878a72\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-87a8253 elementor-widget elementor-widget-accordion\" data-id=\"87a8253\" data-element_type=\"widget\" data-widget_type=\"accordion.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-accordion\">\n\t\t\t\t\t\t\t<div class=\"elementor-accordion-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1421\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1421\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon elementor-accordion-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-closed\"><i class=\"fas fa-book-reader\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-accordion-icon-opened\"><i class=\"fas fa-book-reader\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-accordion-title\" tabindex=\"0\">Separable Differential Equations<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-1421\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1421\"><p style=\"text-align: justify;\"><span style=\"font-size: 18px;\"><strong>Separable differential equations<\/strong> are a fundamental concept in calculus and are widely used in various fields of science and engineering, including physics, biology, and economics. These types of equations involve only first-order derivatives and can be solved by separating the variables and integrating each side with respect to its respective variable. Solving separable differential equations is a powerful tool in understanding dynamic systems and their behavior over time. In this tutorial, we will explore the steps involved in solving separable differential equations using several examples and techniques, including separation of variables and partial fractions.<\/span><\/p><p style=\"text-align: justify;\"><span style=\"font-size: 18px;\">Separable differential equations are arguably the easiest types of differential equations to solve. These types of DEs can be expressed as the product of two functions by separating the variables (hence the name separable differential equations).\u00a0<\/span><span style=\"font-size: 18px;\">A first-order differential equation is said to be a separable differential equation if it can be expressed as follows.<\/span><\/p><p style=\"text-align: center;\"><span style=\"font-size: 18px;\">\\(\\displaystyle\\frac{dy}{dx}=f(x)g(y)\\)<\/span><\/p><p style=\"text-align: justify;\"><span style=\"font-size: 18px;\">Notice that the function \\(f(x)\\) contains only the \\(x\\) terms, whereas \\(g(y)\\) is comprised of only the \\(y\\) terms.<\/span><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4747a34b elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"4747a34b\" data-element_type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8cce7eb elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8cce7eb\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ab46a8e\" data-id=\"ab46a8e\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-dea5846 elementor-widget elementor-widget-presto_video\" data-id=\"dea5846\" 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class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6376085c\" data-id=\"6376085c\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2422ce2 elementor-widget elementor-widget-spacer\" data-id=\"2422ce2\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-dfa134e elementor-align-left elementor-icon-list--layout-traditional elementor-list-item-link-full_width elementor-widget elementor-widget-icon-list\" data-id=\"dfa134e\" data-element_type=\"widget\" data-widget_type=\"icon-list.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul class=\"elementor-icon-list-items\">\n\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<a href=\"https:\/\/onlinemathtutor.co\/index.php\/course-content\/differential-equations-resources\/first-order-de\/problem-set-de-firstorder\/\">\n\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<i aria-hidden=\"true\" class=\"fas fa-arrow-circle-right\"><\/i>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">Back To Main Page: First-Order Differential Equations<\/span>\n\t\t\t\t\t\t\t\t\t\t\t<\/a>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t<\/ul>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-33d7f2ca elementor-widget elementor-widget-spacer\" data-id=\"33d7f2ca\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-spacer\">\n\t\t\t<div class=\"elementor-spacer-inner\"><\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6e77d17a elementor-section-height-min-height elementor-section-content-middle elementor-section-stretched elementor-section-boxed elementor-section-height-default elementor-section-items-middle elementor-invisible\" data-id=\"6e77d17a\" data-element_type=\"section\" 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href=\"https:\/\/onlinemathtutor.co\/course-content\/differential-equations-resources\/numerical-solutions-of-odes\/problem-set-de-numerical\/\" class=\"elementor-sub-item\" tabindex=\"-1\"><img decoding=\"async\" src=\"https:\/\/onlinemathtutor.co\/wp-content\/uploads\/2022\/02\/Problem-Sets.png\"  width=\"20\" style=\"margin-right: 10px;\">  Problem Set<\/a><\/li>\n<\/ul>\n<\/li>\n<\/ul>\t\t\t<\/nav>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-50 elementor-top-column elementor-element elementor-element-21d47f60\" data-id=\"21d47f60\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-580269b3 elementor-widget elementor-widget-spacer\" data-id=\"580269b3\" data-element_type=\"widget\" data-widget_type=\"spacer.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div 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