{"id":1614,"date":"2021-07-22T17:02:44","date_gmt":"2021-07-22T17:02:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=1614"},"modified":"2022-03-21T22:54:24","modified_gmt":"2022-03-21T22:54:24","slug":"solutions-to-differential-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/solutions-to-differential-equations\/","title":{"raw":"Solutions to Differential Equations","rendered":"Solutions to Differential Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\" data-type=\"abstract\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify the order of a differential equation<\/li>\r\n \t<li>Explain what is meant by a solution to a differential equation<\/li>\r\n \t<li>Distinguish between the general solution and a particular solution of a differential equation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section id=\"fs-id1170573512410\" data-depth=\"1\">\r\n<h2 data-type=\"title\">General Differential Equations<\/h2>\r\n<p id=\"fs-id1170573209923\">Consider the equation [latex]{y}^{\\prime }=3{x}^{2}[\/latex], which is an example of a differential equation because it includes a derivative. There is a relationship between the variables [latex]x[\/latex] and [latex]y\\text{:}y[\/latex] is an unknown function of [latex]x[\/latex]. Furthermore, the left-hand side of the equation is the derivative of [latex]y[\/latex]. Therefore we can interpret this equation as follows: Start with some function [latex]y=f\\left(x\\right)[\/latex] and take its derivative. The answer must be equal to [latex]3{x}^{2}[\/latex]. What function has a derivative that is equal to [latex]3{x}^{2}?[\/latex] One such function is [latex]y={x}^{3}[\/latex], so this function is considered a <span data-type=\"term\"><strong>solution to a differential equation<\/strong><\/span><strong>.<\/strong><\/p>\r\n\r\n<div id=\"fs-id1170573431367\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1170571131063\">A <strong>differential equation <\/strong>is an equation involving an unknown function [latex]y=f\\left(x\\right)[\/latex] and one or more of its derivatives. A solution to a differential equation is a function [latex]y=f\\left(x\\right)[\/latex] that satisfies the differential equation when [latex]f[\/latex] and its derivatives are substituted into the equation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170573411846\" class=\"media-2\" data-type=\"note\">\r\n<div class=\"textbox tryit\">\r\n<h3>Interactive<\/h3>\r\n<p id=\"fs-id1170571255388\">Go to <a href=\"https:\/\/demonstrations.wolfram.com\/search.html?query=differential%20equation\" target=\"_blank\" rel=\"noopener\">this website to view demonstrations of differential equations<\/a>.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170573308275\">Some examples of differential equations and their solutions appear in the following table.<\/p>\r\n\r\n<table id=\"fs-id1170573355757\" summary=\"A table with two rows and two columns. The first column has the header \"><caption>\u00a0<\/caption>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-align=\"left\">Equation<\/th>\r\n<th data-align=\"left\">Solution<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]y^{\\prime} =2x[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]y={x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]y^{\\prime} +3y=6x+11[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]y={e}^{-3x}+2x+3[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]y^{\\prime} \\prime -3y^{\\prime} +2y=24{e}^{-2x}[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]y=3{e}^{x}-4{e}^{2x}+2{e}^{-2x}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170573514019\">Note that a solution to a differential equation is not necessarily unique, primarily because the derivative of a constant is zero. For example, [latex]y={x}^{2}+4[\/latex] is also a solution to the first differential equation in the table. We will return to this idea a little bit later in this section. First, we briefly review the rules for derivatives of exponential functions, and then explore what it means for a function to be a solution to a differential equation.<\/p>\r\n\r\n<\/section><section id=\"fs-id1170573512410\" data-depth=\"1\">\r\n<div id=\"fs-id1170573540760\" data-type=\"example\">\r\n<div id=\"fs-id1170573412377\" data-type=\"exercise\">\r\n<div id=\"fs-id1170573734340\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Derivatives of Exponential Functions<\/h3>\r\n<ol>\r\n \t<li>[latex] \\frac{d}{dx} \\left( e^x \\right) = e^x [\/latex]<\/li>\r\n \t<li>[latex] \\frac{d}{dx} \\left( e^{g(x)} \\right) = e^{g(x)}g'(x) [\/latex]\u00a0\u00a0\u00a0 (this is the chain rule applied to the derivative of an exponential function)<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Verifying Solutions of Differential Equations<\/h3>\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<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to Example:\u00a0Verifying Solutions of Differential Equations[\/caption]\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\"><\/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<div id=\"fs-id1170573403872\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1170571153281\" data-type=\"exercise\">\r\n<div id=\"fs-id1170573365858\" data-type=\"problem\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1170573365858\" data-type=\"problem\">\r\n<p id=\"fs-id1170573397796\">Verify that [latex]y=2{e}^{3x}-2x - 2[\/latex] is a solution to the differential equation [latex]{y}^{\\prime }-3y=6x+4[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558898\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558898\"]\r\n<div id=\"fs-id1170573294900\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1170573627896\">First calculate [latex]{y}^{\\prime }[\/latex] then substitute both [latex]{y}^{\\prime }[\/latex] and [latex]y[\/latex] into the left-hand side.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170573366028\">It is convenient to define characteristics of differential equations that make it easier to talk about them and categorize them. The most basic characteristic of a differential equation is its order.<\/p>\r\n\r\n<div id=\"fs-id1170573393994\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1170573394653\">The <strong>order of a differential equation<\/strong> is the highest order of any derivative of the unknown function that appears in the equation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170571085382\" data-type=\"example\">\r\n<div id=\"fs-id1170573394321\" data-type=\"exercise\">\r\n<div id=\"fs-id1170573438001\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Identifying the Order of a Differential Equation<\/h3>\r\n<div id=\"fs-id1170573438001\" data-type=\"problem\">\r\n<p id=\"fs-id1170570999537\">What is the order of each of the following differential equations?<\/p>\r\n\r\n<ol id=\"fs-id1170573386836\" type=\"a\">\r\n \t<li>[latex]{y}^{\\prime }-4y={x}^{2}-3x+4[\/latex]<\/li>\r\n \t<li>[latex]{x}^{2}y\\text{'''}-3xy\\text{''}+x{y}^{\\prime }-3y=\\sin{x}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{4}{x}{y}^{\\left(4\\right)}-\\frac{6}{{x}^{2}}y\\text{''}+\\frac{12}{{x}^{4}}y={x}^{3}-3{x}^{2}+4x - 12[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n[reveal-answer q=\"44558897\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558897\"]\r\n<div id=\"fs-id1170573405146\" data-type=\"solution\">\r\n<ol id=\"fs-id1170573391881\" type=\"a\">\r\n \t<li>The highest derivative in the equation is [latex]{y}^{\\prime }[\/latex], so the order is [latex]1[\/latex].<\/li>\r\n \t<li>The highest derivative in the equation is [latex]y\\text{'''}[\/latex], so the order is [latex]3[\/latex].<\/li>\r\n \t<li>The highest derivative in the equation is [latex]{y}^{\\left(4\\right)}[\/latex], so the order is [latex]4[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to Example:\u00a0Identifying the Order of a Differential Equation[\/caption]\r\n\r\n<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=6722723&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=3BG0hO-FDEw&amp;video_target=tpm-plugin-tevfhbej-3BG0hO-FDEw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/center>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/4.1.1_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for \"4.1.1\" here (opens in new window)<\/a>.\r\n<div id=\"fs-id1170573494361\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1170573290714\" data-type=\"exercise\">\r\n<div id=\"fs-id1170570995771\" data-type=\"problem\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1170570995771\" data-type=\"problem\">\r\n<p id=\"fs-id1170573262020\">What is the order of the following differential equation?<\/p>\r\n\r\n<div id=\"fs-id1170573604103\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\left({x}^{4}-3x\\right){y}^{\\left(5\\right)}-\\left(3{x}^{2}+1\\right){y}^{\\prime }+3y=\\sin{x}\\cos{x}[\/latex]<\/div>\r\n<div data-type=\"equation\" data-label=\"\"><\/div>\r\n<\/div>\r\n[reveal-answer q=\"44558895\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558895\"]\r\n<div id=\"fs-id1170573362785\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1170573389448\">What is the highest derivative in the equation?<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558896\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558896\"]\r\n<div id=\"fs-id1170571141543\" data-type=\"solution\">\r\n<p id=\"fs-id1170570993950\">[latex]5[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1170573365611\" data-depth=\"1\">\r\n<h2 data-type=\"title\">General and Particular Solutions<\/h2>\r\n<p id=\"fs-id1170573587030\">We already noted that the differential equation [latex]{y}^{\\prime }=2x[\/latex] has at least two solutions: [latex]y={x}^{2}[\/latex] and [latex]y={x}^{2}+4[\/latex]. The only difference between these two solutions is the last term, which is a constant. What if the last term is a different constant? Will this expression still be a solution to the differential equation? In fact, any function of the form [latex]y={x}^{2}+C[\/latex], where [latex]C[\/latex] represents any constant, is a solution as well. The reason is that the derivative of [latex]{x}^{2}+C[\/latex] is [latex]2x[\/latex], regardless of the value of [latex]C[\/latex]. It can be shown that any solution of this differential equation must be of the form [latex]y={x}^{2}+C[\/latex]. This is an example of a <strong>general solution<\/strong> to a differential equation. A graph of some of these solutions is given in Figure 1. (<em data-effect=\"italics\">Note<\/em>: in this graph we used even integer values for [latex]C[\/latex] ranging between [latex]-4[\/latex] and [latex]4[\/latex]. In fact, there is no restriction on the value of [latex]C[\/latex]; it can be an integer or not.)<\/p>\r\n\r\n<figure id=\"CNX_Calc_Figure_08_01_001\"><figcaption><\/figcaption>[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11233911\/CNX_Calc_Figure_08_01_001.jpg\" alt=\"A graph of a family of solutions to the differential equation y\u2019 = 2 x, which are of the form y = x ^ 2 + C. Parabolas are drawn for values of C: -4, -2, 0, 2, and 4.\" width=\"325\" height=\"331\" data-media-type=\"image\/jpeg\" \/> Figure 1. Family of solutions to the differential equation [latex]{y}^{\\prime }=2x[\/latex].[\/caption]<\/figure>\r\n<p id=\"fs-id1170571277005\">In this example, we are free to choose any solution we wish; for example, [latex]y={x}^{2}-3[\/latex] is a member of the family of solutions to this differential equation. This is called a <strong>particular solution<\/strong> to the differential equation. A particular solution can often be uniquely identified if we are given additional information about the problem.<\/p>\r\n\r\n<div id=\"fs-id1170573439706\" data-type=\"example\">\r\n<div id=\"fs-id1170573282406\" data-type=\"exercise\">\r\n<div id=\"fs-id1170571254551\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Finding a Particular Solution<\/h3>\r\n<div id=\"fs-id1170571254551\" data-type=\"problem\">\r\n<p id=\"fs-id1170573273508\">Find the particular solution to the differential equation [latex]{y}^{\\prime }=2x[\/latex] passing through the point [latex]\\left(2,7\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558894\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558894\"]\r\n<div id=\"fs-id1170573327628\" data-type=\"solution\">\r\n<p id=\"fs-id1170570994384\">Any function of the form [latex]y={x}^{2}+C[\/latex] is a solution to this differential equation. To determine the value of [latex]C[\/latex], we substitute the values [latex]x=2[\/latex] and [latex]y=7[\/latex] into this equation and solve for [latex]C\\text{:}[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1170573418890\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{}\\\\ \\\\ y={x}^{2}+C\\hfill \\\\ 7={2}^{2}+C=4+C\\hfill \\\\ C=3.\\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170573499195\">Therefore the particular solution passing through the point [latex]\\left(2,7\\right)[\/latex] is [latex]y={x}^{2}+3[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to Example:\u00a0Finding a Particular Solution[\/caption]\r\n\r\n<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=6722724&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Vh1avVTT5Mk&amp;video_target=tpm-plugin-pwh1zkmw-Vh1avVTT5Mk\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/center>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/4.1.3_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for \"4.1.3\" here (opens in new window)<\/a>.\r\n<div id=\"fs-id1170573407571\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1170573243576\" data-type=\"exercise\">\r\n<div id=\"fs-id1170571281440\" data-type=\"problem\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1170571281440\" data-type=\"problem\">\r\n<p id=\"fs-id1170573430997\">Find the particular solution to the differential equation<\/p>\r\n\r\n<div id=\"fs-id1170573310732\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{y}^{\\prime }=4x+3[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170573273901\">passing through the point [latex]\\left(1,7\\right)[\/latex], given that [latex]y=2{x}^{2}+3x+C[\/latex] is a general solution to the differential equation.<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558892\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558892\"]\r\n<div id=\"fs-id1170573399195\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1170573574483\">First substitute [latex]x=1[\/latex] and [latex]y=7[\/latex] into the equation, then solve for [latex]C[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558893\"]\r\n<div id=\"fs-id1170573407290\" data-type=\"solution\">\r\n<p id=\"fs-id1170571048789\">[latex]y=2{x}^{2}+3x+2[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]169306[\/ohm_question]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1170573369171\" data-depth=\"1\"><\/section>","rendered":"<div class=\"textbox learning-objectives\" data-type=\"abstract\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify the order of a differential equation<\/li>\n<li>Explain what is meant by a solution to a differential equation<\/li>\n<li>Distinguish between the general solution and a particular solution of a differential equation<\/li>\n<\/ul>\n<\/div>\n<section id=\"fs-id1170573512410\" data-depth=\"1\">\n<h2 data-type=\"title\">General Differential Equations<\/h2>\n<p id=\"fs-id1170573209923\">Consider the equation [latex]{y}^{\\prime }=3{x}^{2}[\/latex], which is an example of a differential equation because it includes a derivative. There is a relationship between the variables [latex]x[\/latex] and [latex]y\\text{:}y[\/latex] is an unknown function of [latex]x[\/latex]. Furthermore, the left-hand side of the equation is the derivative of [latex]y[\/latex]. Therefore we can interpret this equation as follows: Start with some function [latex]y=f\\left(x\\right)[\/latex] and take its derivative. The answer must be equal to [latex]3{x}^{2}[\/latex]. What function has a derivative that is equal to [latex]3{x}^{2}?[\/latex] One such function is [latex]y={x}^{3}[\/latex], so this function is considered a <span data-type=\"term\"><strong>solution to a differential equation<\/strong><\/span><strong>.<\/strong><\/p>\n<div id=\"fs-id1170573431367\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\n<hr \/>\n<p id=\"fs-id1170571131063\">A <strong>differential equation <\/strong>is an equation involving an unknown function [latex]y=f\\left(x\\right)[\/latex] and one or more of its derivatives. A solution to a differential equation is a function [latex]y=f\\left(x\\right)[\/latex] that satisfies the differential equation when [latex]f[\/latex] and its derivatives are substituted into the equation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170573411846\" class=\"media-2\" data-type=\"note\">\n<div class=\"textbox tryit\">\n<h3>Interactive<\/h3>\n<p id=\"fs-id1170571255388\">Go to <a href=\"https:\/\/demonstrations.wolfram.com\/search.html?query=differential%20equation\" target=\"_blank\" rel=\"noopener\">this website to view demonstrations of differential equations<\/a>.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170573308275\">Some examples of differential equations and their solutions appear in the following table.<\/p>\n<table id=\"fs-id1170573355757\" summary=\"A table with two rows and two columns. The first column has the header\">\n<caption>\u00a0<\/caption>\n<thead>\n<tr valign=\"top\">\n<th data-align=\"left\">Equation<\/th>\n<th data-align=\"left\">Solution<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]y^{\\prime} =2x[\/latex]<\/td>\n<td data-align=\"left\">[latex]y={x}^{2}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]y^{\\prime} +3y=6x+11[\/latex]<\/td>\n<td data-align=\"left\">[latex]y={e}^{-3x}+2x+3[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]y^{\\prime} \\prime -3y^{\\prime} +2y=24{e}^{-2x}[\/latex]<\/td>\n<td data-align=\"left\">[latex]y=3{e}^{x}-4{e}^{2x}+2{e}^{-2x}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170573514019\">Note that a solution to a differential equation is not necessarily unique, primarily because the derivative of a constant is zero. For example, [latex]y={x}^{2}+4[\/latex] is also a solution to the first differential equation in the table. We will return to this idea a little bit later in this section. First, we briefly review the rules for derivatives of exponential functions, and then explore what it means for a function to be a solution to a differential equation.<\/p>\n<\/section>\n<section id=\"fs-id1170573512410\" data-depth=\"1\">\n<div id=\"fs-id1170573540760\" data-type=\"example\">\n<div id=\"fs-id1170573412377\" data-type=\"exercise\">\n<div id=\"fs-id1170573734340\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox examples\">\n<h3>Recall: Derivatives of Exponential Functions<\/h3>\n<ol>\n<li>[latex]\\frac{d}{dx} \\left( e^x \\right) = e^x[\/latex]<\/li>\n<li>[latex]\\frac{d}{dx} \\left( e^{g(x)} \\right) = e^{g(x)}g'(x)[\/latex]\u00a0\u00a0\u00a0 (this is the chain rule applied to the derivative of an exponential function)<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Verifying Solutions of Differential Equations<\/h3>\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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558899\">Show Solution<\/span><\/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<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example:\u00a0Verifying Solutions of Differential Equations<\/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\"><\/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<div id=\"fs-id1170573403872\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1170571153281\" data-type=\"exercise\">\n<div id=\"fs-id1170573365858\" data-type=\"problem\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1170573365858\" data-type=\"problem\">\n<p id=\"fs-id1170573397796\">Verify that [latex]y=2{e}^{3x}-2x - 2[\/latex] is a solution to the differential equation [latex]{y}^{\\prime }-3y=6x+4[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558898\">Hint<\/span><\/p>\n<div id=\"q44558898\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170573294900\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1170573627896\">First calculate [latex]{y}^{\\prime }[\/latex] then substitute both [latex]{y}^{\\prime }[\/latex] and [latex]y[\/latex] into the left-hand side.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170573366028\">It is convenient to define characteristics of differential equations that make it easier to talk about them and categorize them. The most basic characteristic of a differential equation is its order.<\/p>\n<div id=\"fs-id1170573393994\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\n<hr \/>\n<p id=\"fs-id1170573394653\">The <strong>order of a differential equation<\/strong> is the highest order of any derivative of the unknown function that appears in the equation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170571085382\" data-type=\"example\">\n<div id=\"fs-id1170573394321\" data-type=\"exercise\">\n<div id=\"fs-id1170573438001\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Identifying the Order of a Differential Equation<\/h3>\n<div id=\"fs-id1170573438001\" data-type=\"problem\">\n<p id=\"fs-id1170570999537\">What is the order of each of the following differential equations?<\/p>\n<ol id=\"fs-id1170573386836\" type=\"a\">\n<li>[latex]{y}^{\\prime }-4y={x}^{2}-3x+4[\/latex]<\/li>\n<li>[latex]{x}^{2}y\\text{'''}-3xy\\text{''}+x{y}^{\\prime }-3y=\\sin{x}[\/latex]<\/li>\n<li>[latex]\\frac{4}{x}{y}^{\\left(4\\right)}-\\frac{6}{{x}^{2}}y\\text{''}+\\frac{12}{{x}^{4}}y={x}^{3}-3{x}^{2}+4x - 12[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558897\">Show Solution<\/span><\/p>\n<div id=\"q44558897\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170573405146\" data-type=\"solution\">\n<ol id=\"fs-id1170573391881\" type=\"a\">\n<li>The highest derivative in the equation is [latex]{y}^{\\prime }[\/latex], so the order is [latex]1[\/latex].<\/li>\n<li>The highest derivative in the equation is [latex]y\\text{'''}[\/latex], so the order is [latex]3[\/latex].<\/li>\n<li>The highest derivative in the equation is [latex]{y}^{\\left(4\\right)}[\/latex], so the order is [latex]4[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example:\u00a0Identifying the Order of a Differential Equation<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=6722723&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=3BG0hO-FDEw&amp;video_target=tpm-plugin-tevfhbej-3BG0hO-FDEw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/div>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/4.1.1_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for &#8220;4.1.1&#8221; here (opens in new window)<\/a>.<\/p>\n<div id=\"fs-id1170573494361\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1170573290714\" data-type=\"exercise\">\n<div id=\"fs-id1170570995771\" data-type=\"problem\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1170570995771\" data-type=\"problem\">\n<p id=\"fs-id1170573262020\">What is the order of the following differential equation?<\/p>\n<div id=\"fs-id1170573604103\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\left({x}^{4}-3x\\right){y}^{\\left(5\\right)}-\\left(3{x}^{2}+1\\right){y}^{\\prime }+3y=\\sin{x}\\cos{x}[\/latex]<\/div>\n<div data-type=\"equation\" data-label=\"\"><\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558895\">Hint<\/span><\/p>\n<div id=\"q44558895\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170573362785\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1170573389448\">What is the highest derivative in the equation?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558896\">Show Solution<\/span><\/p>\n<div id=\"q44558896\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170571141543\" data-type=\"solution\">\n<p id=\"fs-id1170570993950\">[latex]5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1170573365611\" data-depth=\"1\">\n<h2 data-type=\"title\">General and Particular Solutions<\/h2>\n<p id=\"fs-id1170573587030\">We already noted that the differential equation [latex]{y}^{\\prime }=2x[\/latex] has at least two solutions: [latex]y={x}^{2}[\/latex] and [latex]y={x}^{2}+4[\/latex]. The only difference between these two solutions is the last term, which is a constant. What if the last term is a different constant? Will this expression still be a solution to the differential equation? In fact, any function of the form [latex]y={x}^{2}+C[\/latex], where [latex]C[\/latex] represents any constant, is a solution as well. The reason is that the derivative of [latex]{x}^{2}+C[\/latex] is [latex]2x[\/latex], regardless of the value of [latex]C[\/latex]. It can be shown that any solution of this differential equation must be of the form [latex]y={x}^{2}+C[\/latex]. This is an example of a <strong>general solution<\/strong> to a differential equation. A graph of some of these solutions is given in Figure 1. (<em data-effect=\"italics\">Note<\/em>: in this graph we used even integer values for [latex]C[\/latex] ranging between [latex]-4[\/latex] and [latex]4[\/latex]. In fact, there is no restriction on the value of [latex]C[\/latex]; it can be an integer or not.)<\/p>\n<figure id=\"CNX_Calc_Figure_08_01_001\"><figcaption><\/figcaption><div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11233911\/CNX_Calc_Figure_08_01_001.jpg\" alt=\"A graph of a family of solutions to the differential equation y\u2019 = 2 x, which are of the form y = x ^ 2 + C. Parabolas are drawn for values of C: -4, -2, 0, 2, and 4.\" width=\"325\" height=\"331\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. Family of solutions to the differential equation [latex]{y}^{\\prime }=2x[\/latex].<\/p>\n<\/div>\n<\/figure>\n<p id=\"fs-id1170571277005\">In this example, we are free to choose any solution we wish; for example, [latex]y={x}^{2}-3[\/latex] is a member of the family of solutions to this differential equation. This is called a <strong>particular solution<\/strong> to the differential equation. A particular solution can often be uniquely identified if we are given additional information about the problem.<\/p>\n<div id=\"fs-id1170573439706\" data-type=\"example\">\n<div id=\"fs-id1170573282406\" data-type=\"exercise\">\n<div id=\"fs-id1170571254551\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Finding a Particular Solution<\/h3>\n<div id=\"fs-id1170571254551\" data-type=\"problem\">\n<p id=\"fs-id1170573273508\">Find the particular solution to the differential equation [latex]{y}^{\\prime }=2x[\/latex] passing through the point [latex]\\left(2,7\\right)[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558894\">Show Solution<\/span><\/p>\n<div id=\"q44558894\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170573327628\" data-type=\"solution\">\n<p id=\"fs-id1170570994384\">Any function of the form [latex]y={x}^{2}+C[\/latex] is a solution to this differential equation. To determine the value of [latex]C[\/latex], we substitute the values [latex]x=2[\/latex] and [latex]y=7[\/latex] into this equation and solve for [latex]C\\text{:}[\/latex]<\/p>\n<div id=\"fs-id1170573418890\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{}\\\\ \\\\ y={x}^{2}+C\\hfill \\\\ 7={2}^{2}+C=4+C\\hfill \\\\ C=3.\\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170573499195\">Therefore the particular solution passing through the point [latex]\\left(2,7\\right)[\/latex] is [latex]y={x}^{2}+3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example:\u00a0Finding a Particular Solution<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=6722724&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Vh1avVTT5Mk&amp;video_target=tpm-plugin-pwh1zkmw-Vh1avVTT5Mk\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/div>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/4.1.3_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for &#8220;4.1.3&#8221; here (opens in new window)<\/a>.<\/p>\n<div id=\"fs-id1170573407571\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1170573243576\" data-type=\"exercise\">\n<div id=\"fs-id1170571281440\" data-type=\"problem\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1170571281440\" data-type=\"problem\">\n<p id=\"fs-id1170573430997\">Find the particular solution to the differential equation<\/p>\n<div id=\"fs-id1170573310732\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{y}^{\\prime }=4x+3[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170573273901\">passing through the point [latex]\\left(1,7\\right)[\/latex], given that [latex]y=2{x}^{2}+3x+C[\/latex] is a general solution to the differential equation.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558892\">Hint<\/span><\/p>\n<div id=\"q44558892\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170573399195\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1170573574483\">First substitute [latex]x=1[\/latex] and [latex]y=7[\/latex] into the equation, then solve for [latex]C[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558893\">Show Solution<\/span><\/p>\n<div id=\"q44558893\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170573407290\" data-type=\"solution\">\n<p id=\"fs-id1170571048789\">[latex]y=2{x}^{2}+3x+2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm169306\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=169306&theme=oea&iframe_resize_id=ohm169306&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1170573369171\" data-depth=\"1\"><\/section>\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-1614\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>4.1.2. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>4.1.1. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>4.1.3. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><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":17533,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"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\"},{\"type\":\"original\",\"description\":\"4.1.2\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"4.1.1\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"4.1.3\",\"author\":\"Ryan 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