{"id":23481,"date":"2022-05-28T22:15:00","date_gmt":"2022-05-28T22:15:00","guid":{"rendered":"https:\/\/onlinemathtutor.co\/?page_id=23481"},"modified":"2023-04-21T00:35:56","modified_gmt":"2023-04-21T00:35:56","slug":"practice-problems-de-separable","status":"publish","type":"page","link":"https:\/\/onlinemathtutor.co\/course-content\/differential-equations-resources\/first-order-de\/practice-problems-de-separable\/","title":{"rendered":"Practice Problems &#8211; DE &#8211; [Separable]"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"23481\" class=\"elementor elementor-23481\" data-elementor-post-type=\"page\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-53b6bcce elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"53b6bcce\" 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-1fab8ae0\" data-id=\"1fab8ae0\" 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-629683d elementor-widget elementor-widget-heading\" data-id=\"629683d\" 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\">Practice Problems: Separable DEs<\/h1>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-217d26a2 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=\"217d26a2\" 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-1-217d26a2\" 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-2-217d26a2\" 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\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-5212b996 elementor-section-stretched elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5212b996\" 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-6bc7dc05\" data-id=\"6bc7dc05\" 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-53d81a28 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"53d81a28\" 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-5be2a4e elementor-widget elementor-widget-accordion\" data-id=\"5be2a4e\" 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-9631\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-9631\" 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-9631\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-9631\"><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-2f5ad23a elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"2f5ad23a\" 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-43d832ee elementor-section-stretched elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"43d832ee\" 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-7ba989db\" data-id=\"7ba989db\" 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-72f0aa0f elementor-widget elementor-widget-spacer\" data-id=\"72f0aa0f\" 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-3177e199 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3177e199\" 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-7581812d\" data-id=\"7581812d\" 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-7af153a3 elementor-widget elementor-widget-heading\" data-id=\"7af153a3\" 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<h2 class=\"elementor-heading-title elementor-size-large\">Practice Problems<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-44ad9dc6 elementor-widget elementor-widget-text-editor\" data-id=\"44ad9dc6\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p style=\"text-align: justify;\"><span style=\"font-size: 18px;\">Determine the solution to the given separable differential equation.\u00a0<\/span><\/p>\t\t\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-8af10f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8af10f\" 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-33 elementor-top-column elementor-element elementor-element-43f313a2\" data-id=\"43f313a2\" 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-2ce182de elementor-widget elementor-widget-text-editor\" data-id=\"2ce182de\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ol start=\"1\"><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\frac{dy}{dx}=e^{2x+3y}\\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\frac{dy}{dx}=\\sqrt{y}(x+1)^2\\)<\/li><\/ol>\t\t\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-33 elementor-top-column elementor-element elementor-element-63128105\" data-id=\"63128105\" 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-2e1105de elementor-widget elementor-widget-text-editor\" data-id=\"2e1105de\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ol start=\"3\"><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\frac{dy}{dx}=y^2+1\\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\frac{dy}{dx}=y-y^2\\)<\/li><\/ol>\t\t\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-33 elementor-top-column elementor-element elementor-element-114ac36c\" data-id=\"114ac36c\" 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-700c68cd elementor-widget elementor-widget-text-editor\" data-id=\"700c68cd\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<ol start=\"5\"><li style=\"margin-bottom: 30px;\">\\(\\displaystyle x\\frac{dy}{dx}=2y\\)<\/li><li style=\"margin-bottom: 30px;\">\\(\\displaystyle \\frac{dy}{dx}=\\texttt{cos}(2y)\\texttt{sin}(5x)\\)<\/li><\/ol>\t\t\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-39a2d7a6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"39a2d7a6\" 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-1f606e80\" data-id=\"1f606e80\" 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-74f2c093 elementor-align-left elementor-icon-list--layout-traditional elementor-list-item-link-full_width elementor-widget elementor-widget-icon-list\" data-id=\"74f2c093\" 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\/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 DE Problem Sets<\/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\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-25a4f084 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"25a4f084\" 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-33 elementor-top-column elementor-element elementor-element-1f6356eb\" data-id=\"1f6356eb\" data-element_type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap\">\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-3971b9cb\" data-id=\"3971b9cb\" 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-eda002c elementor-widget elementor-widget-spacer\" data-id=\"eda002c\" 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-5b437bf6 elementor-view-framed elementor-position-left elementor-shape-circle elementor-mobile-position-top elementor-vertical-align-top elementor-widget elementor-widget-icon-box\" data-id=\"5b437bf6\" 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class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-22284\"><a href=\"https:\/\/onlinemathtutor.co\/course-content\/differential-equations-resources\/numerical-solutions-of-odes\/\" class=\"elementor-item\" tabindex=\"-1\">Numerical<\/a>\n<ul class=\"sub-menu elementor-nav-menu--dropdown\">\n\t<li class=\"menu-item menu-item-type-post_type menu-item-object-page menu-item-24025\"><a 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 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