{"id":23349,"date":"2022-05-28T21:27:39","date_gmt":"2022-05-28T21:27:39","guid":{"rendered":"https:\/\/onlinemathtutor.co\/?page_id=23349"},"modified":"2023-04-21T00:38:50","modified_gmt":"2023-04-21T00:38:50","slug":"virtual-lessons-de-intfactor","status":"publish","type":"page","link":"https:\/\/onlinemathtutor.co\/course-content\/differential-equations-resources\/higher-order-de\/practice-problems-de-higherorder\/virtual-lessons-de-intfactor\/","title":{"rendered":"Virtual Lessons &#8211; DE &#8211; [IntFactor]"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"23349\" class=\"elementor elementor-23349\" data-elementor-post-type=\"page\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-19f85e27 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"19f85e27\" 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-186324b3\" data-id=\"186324b3\" 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-381a6c3f elementor-widget elementor-widget-heading\" data-id=\"381a6c3f\" 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: Integrating Factor<\/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-1f889a65 elementor-section-stretched elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1f889a65\" 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-32837b3a\" data-id=\"32837b3a\" 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-535b5800 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=\"535b5800\" 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-1-535b5800\" 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-2-535b5800\" 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-1b70b38d elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"1b70b38d\" 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-44663ac elementor-widget elementor-widget-accordion\" data-id=\"44663ac\" 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-7171\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-7171\" 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\">The Integrating Factor<\/a>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<div id=\"elementor-tab-content-7171\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-7171\"><p style=\"text-align: justify;\"><span style=\"font-size: 18px;\">Differential equations arise in many areas of science and engineering, including physics, chemistry, and biology. Solving differential equations using the integral factor is a powerful technique that can be used to solve first-order linear ordinary differential equations with constant coefficients. This technique involves multiplying both sides of the differential equation by an integrating factor that is derived from the coefficient of the dependent variable. In this tutorial, we will explore the steps involved in solving differential equations using the integral factor, including finding the integrating factor and solving for the general solution.\n<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-size: 18px;\">\nThe general formula for a first-order linear ordinary differential equation with constant coefficients is:\n\\(\\frac{dy}{dx} + p(x)y = q(x)\\) where \\(p(x)\\) and \\(q(x)\\) are continuous functions of \\(x\\). To solve this differential equation using the integral factor, we need to find an integrating factor \\(u(x)=e^{\\int p(x)dx}\\) such that:\n\\(u(x) \\frac{dy}{dx} + p(x)u(x) y = u(x)q(x)\\).\nThis equation can be written in the product rule form \\(\\frac{d}{dx} (u(x)y) = u(x)q(x)\\)\n<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-size: 18px;\">\nBy integrating both sides with respect to \\(x\\), we can solve for <code>\\(u(x)\\)<\/code> and find the general solution to the differential equation. This technique is particularly useful for solving differential equations with non-constant coefficients.\n<\/span><\/p>\n<p style=\"text-align: justify;\"><span style=\"font-size: 18px;\"> <strong><a href=\"https:\/\/onlinemathtutor.co\/index.php\/course-content\/integration\/integration-formulas-update\/\" target=\"_blank\" rel=\"noopener\">See Integral Formulas<\/a><\/strong>.\u00a0<\/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-1c7f580 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"1c7f580\" 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 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