{"id":798,"date":"2025-06-20T17:13:27","date_gmt":"2025-06-20T17:13:27","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=798"},"modified":"2025-07-22T14:24:17","modified_gmt":"2025-07-22T14:24:17","slug":"basics-of-differential-equations-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/basics-of-differential-equations-learn-it-1\/","title":{"raw":"Basics of Differential Equations: Learn It 1","rendered":"Basics of Differential Equations: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Determine the order of a differential equation<\/li>\r\n \t<li>Tell the difference between a general solution and a particular solution<\/li>\r\n \t<li>Identify what makes a problem an initial-value problem<\/li>\r\n \t<li>Check if a function actually solves a given differential equation or initial-value problem<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Basics of Differential Equations<\/h2>\r\n<p class=\"whitespace-normal break-words\">Calculus is the mathematics of change, and we express rates of change using derivatives. This naturally leads us to one of calculus's most powerful applications: <strong>differential equations<\/strong>.<\/p>\r\n<p class=\"whitespace-normal break-words\">A differential equation is simply an equation that contains an unknown function and its derivatives. These equations help us understand how quantities change over time and often reveal the underlying reasons for those changes.<\/p>\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Think of it this way: If you know how fast something is changing (the derivative), can you figure out what the original function was? That's what differential equations help us solve.<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Let's start with a simple example: [latex]y' = 3x^2[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">This equation tells us that we're looking for a function [latex]y = f(x)[\/latex] whose derivative equals [latex]3x^2[\/latex]. In other words:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Start with some unknown function [latex]y = f(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Take its derivative<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The result must equal [latex]3x^2[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">What function has a derivative equal to [latex]3x^2[\/latex]? One answer is [latex]y = x^3[\/latex], since [latex]\\frac{d}{dx}[x^3] = 3x^2[\/latex].<\/p>\r\n\r\n<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>differential equations<\/h3>\r\n<p class=\"whitespace-normal break-words\"><strong>Differential Equation<\/strong>: An equation involving an unknown function [latex]y = f(x)[\/latex] and one or more of its derivatives.<\/p>\r\n[latex]\\\\[\/latex]\r\n<p class=\"whitespace-normal break-words\"><strong>Solution<\/strong>: A solution to a differential equation is a function [latex]y = f(x)[\/latex] that satisfies the differential equation when the function and its derivatives are substituted into the equation.<\/p>\r\n\r\n<\/section>Techniques for solving differential equations vary widely\u2014from direct integration to graphical methods to computer calculations. We'll explore the foundational ideas here and build on them throughout the course.\r\n<p class=\"whitespace-normal break-words\">Here are some examples of differential equations and their solutions:<\/p>\r\n\r\n<table class=\"bg-bg-100 min-w-full border-separate border-spacing-0 text-sm leading-[1.88888] whitespace-normal\">\r\n<thead class=\"border-b-border-100\/50 border-b-[0.5px] text-left\">\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] font-400 px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"text-align: center;\">Differential Equation<\/th>\r\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] font-400 px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"text-align: center;\">Solution<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"text-align: center;\">[latex]y' = 2x[\/latex]<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"text-align: center;\">[latex]y = x^2[\/latex]<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"text-align: center;\">[latex]y' + 3y = 6x + 11[\/latex]<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"text-align: center;\">[latex]y = e^{-3x} + 2x + 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"text-align: center;\">[latex]y'' - 3y' + 2y = 24e^{-2x}[\/latex]<\/td>\r\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"text-align: center;\">[latex]y = 3e^x - 4e^{2x} + 2e^{-2x}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p class=\"whitespace-normal break-words\">Notice how these equations get more complex as we include higher-order derivatives like [latex]y''[\/latex] (the second derivative).<\/p>\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"62\" height=\"55\" \/><strong>Caution\u00a0solutions aren't always unique!<\/strong> A differential equation can have multiple solutions! For example, [latex]y = x^2 + 4[\/latex] is also a solution to [latex]y' = 2x[\/latex] since the derivative of any constant is zero. We'll explore this idea more shortly.<\/section>Before we dive deeper into what makes a function a solution, let's review some key derivative rules you'll need.\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">\r\n<p class=\"whitespace-normal break-words\">Derivatives of Exponential Functions<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(e^x) = e^x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(e^{g(x)}) = e^{g(x)} \\cdot g'(x)[\/latex] (chain rule with exponentials)<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">These rules will be essential as we work with differential equations involving exponential functions.<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1170573734340\" data-type=\"problem\">\r\n<p id=\"fs-id1170573590002\">Verify that the function [latex]y={e}^{-3x}+2x+3[\/latex] is a solution to the differential equation [latex]{y}^{\\prime }+3y=6x+11[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558899\"]\r\n<div id=\"fs-id1170573263434\" data-type=\"solution\">\r\n<p id=\"fs-id1170573410898\">To verify the solution, we first calculate [latex]{y}^{\\prime }[\/latex] using the chain rule for derivatives. This gives [latex]{y}^{\\prime }=-3{e}^{-3x}+2[\/latex]. Next we substitute [latex]y[\/latex] and [latex]{y}^{\\prime }[\/latex] into the left-hand side of the differential equation:<\/p>\r\n\r\n<div id=\"fs-id1170573368104\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\left(-3{e}^{-2x}+2\\right)+3\\left({e}^{-2x}+2x+3\\right)[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170573401064\">The resulting expression can be simplified by first distributing to eliminate the parentheses, giving<\/p>\r\n\r\n<div id=\"fs-id1170573401703\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]-3{e}^{-2x}+2+3{e}^{-2x}+6x+9[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170573582262\">Combining like terms leads to the expression [latex]6x+11[\/latex], which is equal to the right-hand side of the differential equation. This result verifies that [latex]y={e}^{-3x}+2x+3[\/latex] is a solution of the differential equation.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.\r\n\r\n<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=6722722&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Z0PjIYo3Big&amp;video_target=tpm-plugin-84395ywl-Z0PjIYo3Big\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/4.1.2_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for \"4.1.2\" here (opens in new window)<\/a>.\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Determine the order of a differential equation<\/li>\n<li>Tell the difference between a general solution and a particular solution<\/li>\n<li>Identify what makes a problem an initial-value problem<\/li>\n<li>Check if a function actually solves a given differential equation or initial-value problem<\/li>\n<\/ul>\n<\/section>\n<h2>Basics of Differential Equations<\/h2>\n<p class=\"whitespace-normal break-words\">Calculus is the mathematics of change, and we express rates of change using derivatives. This naturally leads us to one of calculus&#8217;s most powerful applications: <strong>differential equations<\/strong>.<\/p>\n<p class=\"whitespace-normal break-words\">A differential equation is simply an equation that contains an unknown function and its derivatives. These equations help us understand how quantities change over time and often reveal the underlying reasons for those changes.<\/p>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Think of it this way: If you know how fast something is changing (the derivative), can you figure out what the original function was? That&#8217;s what differential equations help us solve.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Let&#8217;s start with a simple example: [latex]y' = 3x^2[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">This equation tells us that we&#8217;re looking for a function [latex]y = f(x)[\/latex] whose derivative equals [latex]3x^2[\/latex]. In other words:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Start with some unknown function [latex]y = f(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Take its derivative<\/li>\n<li class=\"whitespace-normal break-words\">The result must equal [latex]3x^2[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">What function has a derivative equal to [latex]3x^2[\/latex]? One answer is [latex]y = x^3[\/latex], since [latex]\\frac{d}{dx}[x^3] = 3x^2[\/latex].<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>differential equations<\/h3>\n<p class=\"whitespace-normal break-words\"><strong>Differential Equation<\/strong>: An equation involving an unknown function [latex]y = f(x)[\/latex] and one or more of its derivatives.<\/p>\n<p>[latex]\\\\[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Solution<\/strong>: A solution to a differential equation is a function [latex]y = f(x)[\/latex] that satisfies the differential equation when the function and its derivatives are substituted into the equation.<\/p>\n<\/section>\n<p>Techniques for solving differential equations vary widely\u2014from direct integration to graphical methods to computer calculations. We&#8217;ll explore the foundational ideas here and build on them throughout the course.<\/p>\n<p class=\"whitespace-normal break-words\">Here are some examples of differential equations and their solutions:<\/p>\n<table class=\"bg-bg-100 min-w-full border-separate border-spacing-0 text-sm leading-[1.88888] whitespace-normal\">\n<thead class=\"border-b-border-100\/50 border-b-[0.5px] text-left\">\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] font-400 px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"text-align: center;\">Differential Equation<\/th>\n<th class=\"text-text-000 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] font-400 px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"text-align: center;\">Solution<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"text-align: center;\">[latex]y' = 2x[\/latex]<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"text-align: center;\">[latex]y = x^2[\/latex]<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"text-align: center;\">[latex]y' + 3y = 6x + 11[\/latex]<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"text-align: center;\">[latex]y = e^{-3x} + 2x + 3[\/latex]<\/td>\n<\/tr>\n<tr class=\"[tbody&gt;&amp;]:odd:bg-bg-500\/10\">\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"text-align: center;\">[latex]y'' - 3y' + 2y = 24e^{-2x}[\/latex]<\/td>\n<td class=\"border-t-border-100\/50 [&amp;:not(:first-child)]:-x-[hsla(var(--border-100) \/ 0.5)] border-t-[0.5px] px-2 [&amp;:not(:first-child)]:border-l-[0.5px]\" style=\"text-align: center;\">[latex]y = 3e^x - 4e^{2x} + 2e^{-2x}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"whitespace-normal break-words\">Notice how these equations get more complex as we include higher-order derivatives like [latex]y''[\/latex] (the second derivative).<\/p>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"62\" height=\"55\" \/><strong>Caution\u00a0solutions aren&#8217;t always unique!<\/strong> A differential equation can have multiple solutions! For example, [latex]y = x^2 + 4[\/latex] is also a solution to [latex]y' = 2x[\/latex] since the derivative of any constant is zero. We&#8217;ll explore this idea more shortly.<\/section>\n<p>Before we dive deeper into what makes a function a solution, let&#8217;s review some key derivative rules you&#8217;ll need.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">\n<p class=\"whitespace-normal break-words\">Derivatives of Exponential Functions<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(e^x) = e^x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\frac{d}{dx}(e^{g(x)}) = e^{g(x)} \\cdot g'(x)[\/latex] (chain rule with exponentials)<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">These rules will be essential as we work with differential equations involving exponential functions.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1170573734340\" data-type=\"problem\">\n<p id=\"fs-id1170573590002\">Verify that the function [latex]y={e}^{-3x}+2x+3[\/latex] is a solution to the differential equation [latex]{y}^{\\prime }+3y=6x+11[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558899\">Show Solution<\/button><\/p>\n<div id=\"q44558899\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170573263434\" data-type=\"solution\">\n<p id=\"fs-id1170573410898\">To verify the solution, we first calculate [latex]{y}^{\\prime }[\/latex] using the chain rule for derivatives. This gives [latex]{y}^{\\prime }=-3{e}^{-3x}+2[\/latex]. Next we substitute [latex]y[\/latex] and [latex]{y}^{\\prime }[\/latex] into the left-hand side of the differential equation:<\/p>\n<div id=\"fs-id1170573368104\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\left(-3{e}^{-2x}+2\\right)+3\\left({e}^{-2x}+2x+3\\right)[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170573401064\">The resulting expression can be simplified by first distributing to eliminate the parentheses, giving<\/p>\n<div id=\"fs-id1170573401703\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]-3{e}^{-2x}+2+3{e}^{-2x}+6x+9[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170573582262\">Combining like terms leads to the expression [latex]6x+11[\/latex], which is equal to the right-hand side of the differential equation. This result verifies that [latex]y={e}^{-3x}+2x+3[\/latex] is a solution of the differential equation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=6722722&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Z0PjIYo3Big&amp;video_target=tpm-plugin-84395ywl-Z0PjIYo3Big\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/4.1.2_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for &#8220;4.1.2&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":6,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":669,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/798"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":7,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/798\/revisions"}],"predecessor-version":[{"id":1372,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/798\/revisions\/1372"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/669"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/798\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=798"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=798"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=798"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=798"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}