{"id":814,"date":"2025-06-20T17:15:19","date_gmt":"2025-06-20T17:15:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=814"},"modified":"2025-09-09T18:54:29","modified_gmt":"2025-09-09T18:54:29","slug":"direction-fields-and-eulers-method-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/direction-fields-and-eulers-method-fresh-take\/","title":{"raw":"Direction Fields and Euler's Method: Fresh Take","rendered":"Direction Fields and Euler&#8217;s Method: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Create direction fields for first-order differential equations<\/li>\r\n \t<li>Use a direction field to sketch solution curves<\/li>\r\n \t<li>Use Euler's Method to find approximate solutions step by step<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Direction Fields<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Think of direction fields as a roadmap that shows you which way to go at every intersection\u2014except instead of roads, you're following the flow of solution curves through a differential equation.<\/p>\r\n<p class=\"whitespace-normal break-words\">At any point [latex](x, y)[\/latex], the differential equation [latex]y' = f(x,y)[\/latex] tells you the slope a solution curve must have if it passes through that point. A direction field shows these slopes as tiny arrows scattered across the plane.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>How It Works:<\/strong><\/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\">Pick any point [latex](x_0, y_0)[\/latex] on the coordinate plane<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plug these values into the right side of your equation: [latex]y' = f(x_0, y_0)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Draw a small line segment at that point with the calculated slope<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Repeat for lots of points to see the overall pattern<\/li>\r\n<\/ul>\r\nSolution curves must flow along the direction field like a river follows the landscape. The arrows show the \"current\" that carries your solution forward.\r\n\r\n<\/div>\r\n<h2 data-type=\"title\">Using Direction Fields<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Think of using a direction field like following a hiking trail where each arrow points you toward the next step. You don't need to solve the equation\u2014just follow the visual clues to see where solutions go.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy:<\/strong><\/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\"><strong>Start anywhere:<\/strong> Pick an initial point [latex](x_0, y_0)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Check the slope:<\/strong> Use the direction field arrow at that point<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Take a small step:<\/strong> Move slightly in the direction the arrow points<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Repeat:<\/strong> At your new location, check the new arrow and continue<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Connect the dots:<\/strong> The path you trace is your solution curve<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Each tiny step uses the formula [latex]\\Delta y \\approx y' \\cdot \\Delta x[\/latex]. You're essentially building a solution curve one small linear piece at a time.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Key Insights from Direction Fields:<\/strong><\/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\"><strong>Equilibrium lines:<\/strong> Where arrows are horizontal ([latex]y' = 0[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Solution behavior:<\/strong> Do curves spread apart, come together, or cycle?<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Long-term trends:<\/strong> Where do solutions end up as [latex]x \\to \\infty[\/latex]?<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox interact\" aria-label=\"Interact\">Visit\u00a0<a href=\"https:\/\/www.mathopenref.com\/calcslopefields.html\" target=\"_blank\" rel=\"noopener\">this Java applet for more practice with slope fields<\/a>.<\/section>\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Equilibrium Solutions and Their Stability<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Think of equilibrium solutions as the \"steady states\" where a system naturally wants to settle\u2014like a ball coming to rest at the bottom of a hill or your coffee cooling to room temperature.<\/p>\r\n<p class=\"whitespace-normal break-words\">Equilibrium solutions are constant solutions where [latex]y' = 0[\/latex] everywhere. If [latex]y = k[\/latex] is an equilibrium, then substituting this constant into your differential equation gives zero.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Finding Equilibrium Solutions:<\/strong><\/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\">Set the right-hand side of [latex]y' = f(x,y)[\/latex] equal to zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve for constant values: [latex]f(x,k) = 0[\/latex] for all [latex]x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">These [latex]k[\/latex] values are your equilibrium solutions<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>The Three Types of Stability:<\/strong><\/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\"><strong>Stable:<\/strong> Solutions near the equilibrium get pulled toward it\u2014like a marble rolling toward the bottom of a bowl<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Unstable:<\/strong> Solutions near the equilibrium get pushed away\u2014like balancing a pencil on its tip<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Semi-stable:<\/strong> Mixed behavior\u2014stable from one side, unstable from the other<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Stability Check:<\/strong> Look at arrows just above and below each equilibrium:<\/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\">Arrows pointing toward the equilibrium = <strong>Stable<\/strong><\/li>\r\n \t<li class=\"whitespace-normal break-words\">Arrows pointing away from the equilibrium = <strong>Unstable<\/strong><\/li>\r\n \t<li class=\"whitespace-normal break-words\">Mixed directions = <strong>Semi-stable<\/strong><\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1170571290427\" data-type=\"problem\">\r\n<p id=\"fs-id1170573519611\">Create a direction field for the differential equation [latex]y^{\\prime} =\\left(x+5\\right)\\left(y+2\\right)\\left({y}^{2}-4y+4\\right)[\/latex] and identify any equilibrium solutions. Classify each of the equilibrium solutions as stable, unstable, or semi-stable.<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558895\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558895\"]\r\n<div id=\"fs-id1170571027317\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1170571332593\">First create the direction field and look for horizontal dashes that go all the way across. Then examine the slope lines directly above and below the equilibrium solutions.<\/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-id1170573255134\" data-type=\"solution\">\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11233955\/CNX_Calc_Figure_08_02_008.jpg\" alt=\"A direction field with arrows pointing to the right at y = -4 and y = 4. The arrows point up for y &gt; -4 and down for y &lt; -4. Close to y = 4, the arrows are more horizontal, but the further away, the more vertical they become.\" width=\"487\" height=\"445\" \/>\r\n\r\nThe equilibrium solutions are [latex]y=-2[\/latex] and [latex]y=2[\/latex]. For this equation, [latex]y=-2[\/latex] is an unstable equilibrium solution, and [latex]y=2[\/latex] is a semi-stable equilibrium solution.\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Euler\u2019s Method<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Think of Euler's Method like following GPS directions\u2014instead of knowing the entire route upfront, you get turn-by-turn instructions based on where you are right now and which direction you should head next.<\/p>\r\n<p class=\"whitespace-normal break-words\">When you can't solve a differential equation exactly, use the slope information to build an approximate solution by taking lots of small, straight-line steps.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy:<\/strong><\/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 at your initial point [latex](x_0, y_0)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate the slope using [latex]y' = f(x_0, y_0)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Take a small step of size [latex]h[\/latex] in that direction<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Land at a new point, recalculate the slope, repeat<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Key Formulas:<\/strong><\/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\">[latex]x_n = x_0 + nh[\/latex] (where you are horizontally after [latex]n[\/latex] steps)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y_n = y_{n-1} + h \\cdot f(x_{n-1}, y_{n-1})[\/latex] (your new [latex]y[\/latex]-value)<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Smaller [latex]h[\/latex] values give better accuracy but require more work. Think of it like resolution\u2014higher resolution (smaller [latex]h[\/latex]) gives a clearer picture but takes longer to process.<\/p>\r\nEuler's Method gives approximations, not exact answers. It's like sketching a curve with straight line segments\u2014the more segments you use, the smoother your approximation becomes.\r\n\r\n<\/div>\r\n<section class=\"textbox interact\" aria-label=\"Interact\">Visit <a href=\"https:\/\/www.geogebra.org\/m\/KNxfhNmq\" target=\"_blank\" rel=\"noopener\">this applet for more practice using Euler\u2019s method<\/a>.<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1170573756807\" data-type=\"problem\">\r\n<p id=\"fs-id1170571103641\">Consider the initial-value problem<\/p>\r\n\r\n<div id=\"fs-id1170571103644\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{y}^{\\prime }=3{x}^{2}-{y}^{2}+1,y\\left(0\\right)=2[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170571103620\">Use Euler\u2019s method with a step size of [latex]0.1[\/latex] to generate a table of values for the solution for values of [latex]x[\/latex] between [latex]0[\/latex] and [latex]1[\/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-id1170570999104\" data-type=\"solution\">\r\n<p id=\"fs-id1170570999107\">We are given [latex]h=0.1[\/latex] and [latex]f\\left(x,y\\right)=3{x}^{2}-{y}^{2}+1[\/latex]. Furthermore, the initial condition [latex]y\\left(0\\right)=2[\/latex] gives [latex]{x}_{0}=0[\/latex] and [latex]{y}_{0}=2[\/latex]. Using Euler's method with [latex]n=0[\/latex], we can generate the following table.<\/p>\r\n\r\n<table id=\"fs-id1170571140163\" summary=\"A table with three columns and twelve rows. The first column has the header \"><caption><span data-type=\"title\">Using Euler\u2019s Method to Approximate Solutions to a Differential Equation<\/span><\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]n[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]{x}_{n}[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]{y}_{n}={y}_{n - 1}+hf\\left({x}_{n - 1},{y}_{n - 1}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]1[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]0.1[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]{y}_{1}={y}_{0}+hf\\left({x}_{0},{y}_{0}\\right)=1.7[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]2[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]0.2[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]{y}_{2}={y}_{1}+hf\\left({x}_{1},{y}_{1}\\right)=1.514[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]3[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]0.3[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]{y}_{3}={y}_{2}+hf\\left({x}_{2},{y}_{2}\\right)=1.3968[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]4[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]0.4[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]{y}_{4}={y}_{3}+hf\\left({x}_{3},{y}_{3}\\right)=1.3287[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]5[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]0.5[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]{y}_{5}={y}_{4}+hf\\left({x}_{4},{y}_{4}\\right)=1.3001[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]6[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]0.6[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]{y}_{6}={y}_{5}+hf\\left({x}_{5},{y}_{5}\\right)=1.3061[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]7[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]0.7[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]{y}_{7}={y}_{6}+hf\\left({x}_{6},{y}_{6}\\right)=1.3435[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]8[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]0.8[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]{y}_{8}={y}_{7}+hf\\left({x}_{7},{y}_{7}\\right)=1.4100[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]9[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]0.9[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]{y}_{9}={y}_{8}+hf\\left({x}_{8},{y}_{8}\\right)=1.5032[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">[latex]10[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]1.0[\/latex]<\/td>\r\n<td data-align=\"left\">[latex]{y}_{10}={y}_{9}+hf\\left({x}_{9},{y}_{9}\\right)=1.6202[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170571086078\">With ten calculations, we are able to approximate the values of the solution to the initial-value problem for values of [latex]x[\/latex] between [latex]0[\/latex] and [latex]1[\/latex].<\/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 videos to see the worked solution to the above example.<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/D9YQRFYG_XU?controls=0&amp;start=59&amp;end=425&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/4.2.3_59to425_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"4.2.3\" here (opens in new window)<\/a>.\r\n\r\n<\/section><section class=\"textbox interact\" aria-label=\"Interact\">Visit <a href=\"https:\/\/www.maa.org\/press\/periodicals\/loci\/joma\/the-sir-model-for-spread-of-disease\" target=\"_blank\" rel=\"noopener\">this website for a practical application of differential equations<\/a>.<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Create direction fields for first-order differential equations<\/li>\n<li>Use a direction field to sketch solution curves<\/li>\n<li>Use Euler&#8217;s Method to find approximate solutions step by step<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Direction Fields<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Think of direction fields as a roadmap that shows you which way to go at every intersection\u2014except instead of roads, you&#8217;re following the flow of solution curves through a differential equation.<\/p>\n<p class=\"whitespace-normal break-words\">At any point [latex](x, y)[\/latex], the differential equation [latex]y' = f(x,y)[\/latex] tells you the slope a solution curve must have if it passes through that point. A direction field shows these slopes as tiny arrows scattered across the plane.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>How It Works:<\/strong><\/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\">Pick any point [latex](x_0, y_0)[\/latex] on the coordinate plane<\/li>\n<li class=\"whitespace-normal break-words\">Plug these values into the right side of your equation: [latex]y' = f(x_0, y_0)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Draw a small line segment at that point with the calculated slope<\/li>\n<li class=\"whitespace-normal break-words\">Repeat for lots of points to see the overall pattern<\/li>\n<\/ul>\n<p>Solution curves must flow along the direction field like a river follows the landscape. The arrows show the &#8220;current&#8221; that carries your solution forward.<\/p>\n<\/div>\n<h2 data-type=\"title\">Using Direction Fields<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Think of using a direction field like following a hiking trail where each arrow points you toward the next step. You don&#8217;t need to solve the equation\u2014just follow the visual clues to see where solutions go.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy:<\/strong><\/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\"><strong>Start anywhere:<\/strong> Pick an initial point [latex](x_0, y_0)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Check the slope:<\/strong> Use the direction field arrow at that point<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Take a small step:<\/strong> Move slightly in the direction the arrow points<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Repeat:<\/strong> At your new location, check the new arrow and continue<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Connect the dots:<\/strong> The path you trace is your solution curve<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Each tiny step uses the formula [latex]\\Delta y \\approx y' \\cdot \\Delta x[\/latex]. You&#8217;re essentially building a solution curve one small linear piece at a time.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Key Insights from Direction Fields:<\/strong><\/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\"><strong>Equilibrium lines:<\/strong> Where arrows are horizontal ([latex]y' = 0[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Solution behavior:<\/strong> Do curves spread apart, come together, or cycle?<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Long-term trends:<\/strong> Where do solutions end up as [latex]x \\to \\infty[\/latex]?<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox interact\" aria-label=\"Interact\">Visit\u00a0<a href=\"https:\/\/www.mathopenref.com\/calcslopefields.html\" target=\"_blank\" rel=\"noopener\">this Java applet for more practice with slope fields<\/a>.<\/section>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Equilibrium Solutions and Their Stability<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Think of equilibrium solutions as the &#8220;steady states&#8221; where a system naturally wants to settle\u2014like a ball coming to rest at the bottom of a hill or your coffee cooling to room temperature.<\/p>\n<p class=\"whitespace-normal break-words\">Equilibrium solutions are constant solutions where [latex]y' = 0[\/latex] everywhere. If [latex]y = k[\/latex] is an equilibrium, then substituting this constant into your differential equation gives zero.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Finding Equilibrium Solutions:<\/strong><\/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\">Set the right-hand side of [latex]y' = f(x,y)[\/latex] equal to zero<\/li>\n<li class=\"whitespace-normal break-words\">Solve for constant values: [latex]f(x,k) = 0[\/latex] for all [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">These [latex]k[\/latex] values are your equilibrium solutions<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>The Three Types of Stability:<\/strong><\/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\"><strong>Stable:<\/strong> Solutions near the equilibrium get pulled toward it\u2014like a marble rolling toward the bottom of a bowl<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Unstable:<\/strong> Solutions near the equilibrium get pushed away\u2014like balancing a pencil on its tip<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Semi-stable:<\/strong> Mixed behavior\u2014stable from one side, unstable from the other<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Stability Check:<\/strong> Look at arrows just above and below each equilibrium:<\/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\">Arrows pointing toward the equilibrium = <strong>Stable<\/strong><\/li>\n<li class=\"whitespace-normal break-words\">Arrows pointing away from the equilibrium = <strong>Unstable<\/strong><\/li>\n<li class=\"whitespace-normal break-words\">Mixed directions = <strong>Semi-stable<\/strong><\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1170571290427\" data-type=\"problem\">\n<p id=\"fs-id1170573519611\">Create a direction field for the differential equation [latex]y^{\\prime} =\\left(x+5\\right)\\left(y+2\\right)\\left({y}^{2}-4y+4\\right)[\/latex] and identify any equilibrium solutions. Classify each of the equilibrium solutions as stable, unstable, or semi-stable.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558895\">Hint<\/button><\/p>\n<div id=\"q44558895\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170571027317\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1170571332593\">First create the direction field and look for horizontal dashes that go all the way across. Then examine the slope lines directly above and below the equilibrium solutions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558896\">Show Solution<\/button><\/p>\n<div id=\"q44558896\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170573255134\" data-type=\"solution\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11233955\/CNX_Calc_Figure_08_02_008.jpg\" alt=\"A direction field with arrows pointing to the right at y = -4 and y = 4. The arrows point up for y &gt; -4 and down for y &lt; -4. Close to y = 4, the arrows are more horizontal, but the further away, the more vertical they become.\" width=\"487\" height=\"445\" \/><\/p>\n<p>The equilibrium solutions are [latex]y=-2[\/latex] and [latex]y=2[\/latex]. For this equation, [latex]y=-2[\/latex] is an unstable equilibrium solution, and [latex]y=2[\/latex] is a semi-stable equilibrium solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2>Euler\u2019s Method<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Think of Euler&#8217;s Method like following GPS directions\u2014instead of knowing the entire route upfront, you get turn-by-turn instructions based on where you are right now and which direction you should head next.<\/p>\n<p class=\"whitespace-normal break-words\">When you can&#8217;t solve a differential equation exactly, use the slope information to build an approximate solution by taking lots of small, straight-line steps.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy:<\/strong><\/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 at your initial point [latex](x_0, y_0)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Calculate the slope using [latex]y' = f(x_0, y_0)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Take a small step of size [latex]h[\/latex] in that direction<\/li>\n<li class=\"whitespace-normal break-words\">Land at a new point, recalculate the slope, repeat<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Key Formulas:<\/strong><\/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\">[latex]x_n = x_0 + nh[\/latex] (where you are horizontally after [latex]n[\/latex] steps)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y_n = y_{n-1} + h \\cdot f(x_{n-1}, y_{n-1})[\/latex] (your new [latex]y[\/latex]-value)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Smaller [latex]h[\/latex] values give better accuracy but require more work. Think of it like resolution\u2014higher resolution (smaller [latex]h[\/latex]) gives a clearer picture but takes longer to process.<\/p>\n<p>Euler&#8217;s Method gives approximations, not exact answers. It&#8217;s like sketching a curve with straight line segments\u2014the more segments you use, the smoother your approximation becomes.<\/p>\n<\/div>\n<section class=\"textbox interact\" aria-label=\"Interact\">Visit <a href=\"https:\/\/www.geogebra.org\/m\/KNxfhNmq\" target=\"_blank\" rel=\"noopener\">this applet for more practice using Euler\u2019s method<\/a>.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1170573756807\" data-type=\"problem\">\n<p id=\"fs-id1170571103641\">Consider the initial-value problem<\/p>\n<div id=\"fs-id1170571103644\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{y}^{\\prime }=3{x}^{2}-{y}^{2}+1,y\\left(0\\right)=2[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170571103620\">Use Euler\u2019s method with a step size of [latex]0.1[\/latex] to generate a table of values for the solution for values of [latex]x[\/latex] between [latex]0[\/latex] and [latex]1[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558894\">Show Solution<\/button><\/p>\n<div id=\"q44558894\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170570999104\" data-type=\"solution\">\n<p id=\"fs-id1170570999107\">We are given [latex]h=0.1[\/latex] and [latex]f\\left(x,y\\right)=3{x}^{2}-{y}^{2}+1[\/latex]. Furthermore, the initial condition [latex]y\\left(0\\right)=2[\/latex] gives [latex]{x}_{0}=0[\/latex] and [latex]{y}_{0}=2[\/latex]. Using Euler&#8217;s method with [latex]n=0[\/latex], we can generate the following table.<\/p>\n<table id=\"fs-id1170571140163\" summary=\"A table with three columns and twelve rows. The first column has the header\">\n<caption><span data-type=\"title\">Using Euler\u2019s Method to Approximate Solutions to a Differential Equation<\/span><\/caption>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]n[\/latex]<\/td>\n<td data-align=\"left\">[latex]{x}_{n}[\/latex]<\/td>\n<td data-align=\"left\">[latex]{y}_{n}={y}_{n - 1}+hf\\left({x}_{n - 1},{y}_{n - 1}\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\n<td data-align=\"left\">[latex]0[\/latex]<\/td>\n<td data-align=\"left\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]1[\/latex]<\/td>\n<td data-align=\"left\">[latex]0.1[\/latex]<\/td>\n<td data-align=\"left\">[latex]{y}_{1}={y}_{0}+hf\\left({x}_{0},{y}_{0}\\right)=1.7[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]2[\/latex]<\/td>\n<td data-align=\"left\">[latex]0.2[\/latex]<\/td>\n<td data-align=\"left\">[latex]{y}_{2}={y}_{1}+hf\\left({x}_{1},{y}_{1}\\right)=1.514[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]3[\/latex]<\/td>\n<td data-align=\"left\">[latex]0.3[\/latex]<\/td>\n<td data-align=\"left\">[latex]{y}_{3}={y}_{2}+hf\\left({x}_{2},{y}_{2}\\right)=1.3968[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]4[\/latex]<\/td>\n<td data-align=\"left\">[latex]0.4[\/latex]<\/td>\n<td data-align=\"left\">[latex]{y}_{4}={y}_{3}+hf\\left({x}_{3},{y}_{3}\\right)=1.3287[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]5[\/latex]<\/td>\n<td data-align=\"left\">[latex]0.5[\/latex]<\/td>\n<td data-align=\"left\">[latex]{y}_{5}={y}_{4}+hf\\left({x}_{4},{y}_{4}\\right)=1.3001[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]6[\/latex]<\/td>\n<td data-align=\"left\">[latex]0.6[\/latex]<\/td>\n<td data-align=\"left\">[latex]{y}_{6}={y}_{5}+hf\\left({x}_{5},{y}_{5}\\right)=1.3061[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]7[\/latex]<\/td>\n<td data-align=\"left\">[latex]0.7[\/latex]<\/td>\n<td data-align=\"left\">[latex]{y}_{7}={y}_{6}+hf\\left({x}_{6},{y}_{6}\\right)=1.3435[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]8[\/latex]<\/td>\n<td data-align=\"left\">[latex]0.8[\/latex]<\/td>\n<td data-align=\"left\">[latex]{y}_{8}={y}_{7}+hf\\left({x}_{7},{y}_{7}\\right)=1.4100[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]9[\/latex]<\/td>\n<td data-align=\"left\">[latex]0.9[\/latex]<\/td>\n<td data-align=\"left\">[latex]{y}_{9}={y}_{8}+hf\\left({x}_{8},{y}_{8}\\right)=1.5032[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">[latex]10[\/latex]<\/td>\n<td data-align=\"left\">[latex]1.0[\/latex]<\/td>\n<td data-align=\"left\">[latex]{y}_{10}={y}_{9}+hf\\left({x}_{9},{y}_{9}\\right)=1.6202[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170571086078\">With ten calculations, we are able to approximate the values of the solution to the initial-value problem for values of [latex]x[\/latex] between [latex]0[\/latex] and [latex]1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following videos to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/D9YQRFYG_XU?controls=0&amp;start=59&amp;end=425&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/4.2.3_59to425_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;4.2.3&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n<section class=\"textbox interact\" aria-label=\"Interact\">Visit <a href=\"https:\/\/www.maa.org\/press\/periodicals\/loci\/joma\/the-sir-model-for-spread-of-disease\" target=\"_blank\" rel=\"noopener\">this website for a practical application of differential equations<\/a>.<\/section>\n","protected":false},"author":15,"menu_order":17,"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\/814"}],"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\/814\/revisions"}],"predecessor-version":[{"id":2247,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/814\/revisions\/2247"}],"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\/814\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=814"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=814"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=814"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=814"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}