{"id":793,"date":"2025-06-20T17:13:42","date_gmt":"2025-06-20T17:13:42","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=793"},"modified":"2025-09-08T18:50:08","modified_gmt":"2025-09-08T18:50:08","slug":"basics-of-differential-equations-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/basics-of-differential-equations-fresh-take\/","title":{"raw":"Basics of Differential Equations: Fresh Take","rendered":"Basics of Differential Equations: Fresh Take"},"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<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Think of differential equations as detective work\u2014you're given clues about how something changes, and your job is to figure out what that \"something\" actually is.<\/p>\r\n<p class=\"whitespace-normal break-words\">A differential equation contains an unknown function and its derivatives. The solution is the actual function that makes the equation true.<\/p>\r\n<p class=\"whitespace-normal break-words\">You're told \"the speed of a car is always 60 mph.\" From this rate, you can figure out the car's position: [latex]y = 60t + C[\/latex]. That's differential equations!<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Key Points:<\/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>Input:<\/strong> Rate of change information (the derivative)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Output:<\/strong> The original function we're hunting for<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Reality check:<\/strong> Take your answer's derivative and plug it back in\u2014both sides should match<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Solutions aren't unique! Most differential equations have infinite solutions that differ by a constant (+C).<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Verification Strategy:<\/strong><\/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\">Take the derivative of your proposed solution<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute both the function and derivative into the original equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify\u2014if both sides match, you're golden<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\">These equations model real-world change\u2014population growth, cooling coffee, stock trends. You're learning to work backwards from \"how it changes\" to \"what it actually is.\"<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox interact\" aria-label=\"Interact\">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>.<\/section><section class=\"textbox example\" aria-label=\"Example\">\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<\/section>\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Order of Differential Equations<\/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 the \"order\" of a differential equation like ranking difficulty levels in a video game\u2014the higher the order, the more complex the equation becomes.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Simple Rule:<\/strong> The order equals the highest derivative that appears anywhere in the equation. That's it!<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Quick Examples:<\/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]y' = 2x[\/latex] \u2192 <strong>First-order<\/strong> (highest derivative is [latex]y'[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y'' - 3y' + 2y = 0[\/latex] \u2192 <strong>Second-order<\/strong> (highest derivative is [latex]y''[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y''' + xy' = \\sin(x)[\/latex] \u2192 <strong>Third-order<\/strong> (highest derivative is [latex]y'''[\/latex])<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Common Notation to Watch For:<\/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]y'[\/latex], [latex]y''[\/latex], [latex]y'''[\/latex] (prime notation)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y^{(4)}[\/latex], [latex]y^{(5)}[\/latex] (parentheses for higher orders)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Sometimes written as [latex]\\frac{dy}{dx}[\/latex], [latex]\\frac{d^2y}{dx^2}[\/latex], etc.<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Why Order Matters:<\/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>First-order:<\/strong> Usually the most straightforward to solve<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Second-order:<\/strong> Common in physics (think springs, pendulums)<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Higher-order:<\/strong> More complex, often requiring specialized techniques<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Don't get distracted by complicated-looking coefficients or messy right-hand sides. Just scan the equation for the highest derivative and you've got your answer.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\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<\/section>\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">General Solutions vs. Particular Solutions<\/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 general solutions like a recipe that says \"add some salt to taste\"\u2014there's flexibility built in. A particular solution is like following that recipe and adding exactly 2 teaspoons of salt.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Key Distinction:<\/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>General Solution:<\/strong> Contains arbitrary constants (like [latex]C[\/latex]) and represents an entire family of functions<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Particular Solution:<\/strong> Has specific values for all constants\u2014just one function from the family<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Why does this happen? When you take derivatives, constants disappear. So when you work backwards from a derivative to find the original function, you need to account for any constant that could have been there originally.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Example:<\/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\">For [latex]y' = 2x[\/latex]:\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\">General: [latex]y = x^2 + C[\/latex] (infinite solutions)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Particular: [latex]y = x^2 + 3[\/latex] (one specific solution when [latex]C = 3[\/latex])<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\"><strong>Finding Particular Solutions:<\/strong><\/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\">Start with the general solution<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use given information (like a point the curve passes through)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute and solve for the constant(s)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Replace [latex]C[\/latex] with the specific value<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\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<\/section>\r\n<h2>Initial-Value Problems<\/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\">What's an Initial-Value Problem? It's a differential equation plus one or more <strong>initial conditions<\/strong> that tell you specific values at a particular point (usually when [latex]t = 0[\/latex]).<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Magic Number Rule:<\/strong> You need exactly as many initial conditions as the order of your differential equation:<\/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>First-order equation<\/strong> \u2192 Need 1 initial condition<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Second-order equation<\/strong> \u2192 Need 2 initial conditions<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Third-order equation<\/strong> \u2192 Need 3 initial conditions<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">Why \"Initial\" Values? The independent variable often represents time, so [latex]t = 0[\/latex] is your starting point\u2014like knowing where a ball starts before tracking its motion.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy:<\/strong><\/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\"><strong>Solve the differential equation<\/strong> \u2192 Get the general solution (with constants like [latex]C[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Apply the initial condition(s)<\/strong> \u2192 Substitute given values to find the specific constants<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\">Your solution must satisfy both the differential equation AND the initial condition. Test both separately!<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1170573365823\" data-type=\"problem\">\r\n<p id=\"fs-id1170573429527\">Verify that [latex]y=3{e}^{2t}+4\\sin{t}[\/latex] is a solution to the initial-value problem<\/p>\r\n\r\n<div id=\"fs-id1170573368474\" class=\"unnumbered\" style=\"text-align: left;\" data-type=\"equation\" data-label=\"\">[latex]{y}^{\\prime }-2y=4\\cos{t} - 8\\sin{t},y\\left(0\\right)=3[\/latex].<\/div>\r\n<\/div>\r\n[reveal-answer q=\"44558890\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558890\"]\r\n<div id=\"fs-id1170571276500\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1170573338054\">First verify that [latex]y[\/latex] solves the differential equation. Then check the initial value.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1170573362845\" data-type=\"problem\">\r\n<p id=\"fs-id1170573362847\">Solve the initial-value problem<\/p>\r\n\r\n<div id=\"fs-id1170573570394\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{y}^{\\prime }={x}^{2}-4x+3 - 6{e}^{x},y\\left(0\\right)=8[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558869\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558869\"]\r\n<div id=\"fs-id1170571418500\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1170573502068\">First take the antiderivative of both sides of the differential equation. Then substitute [latex]x=0[\/latex] and [latex]y=8[\/latex] into the resulting equation and solve for [latex]C[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558879\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558879\"]\r\n<div id=\"fs-id1170571455482\" data-type=\"solution\">\r\n<p id=\"fs-id1170571203542\">[latex]y=\\frac{1}{3}{x}^{3}-2{x}^{2}+3x - 6{e}^{x}+14[\/latex]<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\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<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Think of differential equations as detective work\u2014you&#8217;re given clues about how something changes, and your job is to figure out what that &#8220;something&#8221; actually is.<\/p>\n<p class=\"whitespace-normal break-words\">A differential equation contains an unknown function and its derivatives. The solution is the actual function that makes the equation true.<\/p>\n<p class=\"whitespace-normal break-words\">You&#8217;re told &#8220;the speed of a car is always 60 mph.&#8221; From this rate, you can figure out the car&#8217;s position: [latex]y = 60t + C[\/latex]. That&#8217;s differential equations!<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Key Points:<\/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>Input:<\/strong> Rate of change information (the derivative)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Output:<\/strong> The original function we&#8217;re hunting for<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Reality check:<\/strong> Take your answer&#8217;s derivative and plug it back in\u2014both sides should match<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Solutions aren&#8217;t unique! Most differential equations have infinite solutions that differ by a constant (+C).<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Verification Strategy:<\/strong><\/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\">Take the derivative of your proposed solution<\/li>\n<li class=\"whitespace-normal break-words\">Substitute both the function and derivative into the original equation<\/li>\n<li class=\"whitespace-normal break-words\">Simplify\u2014if both sides match, you&#8217;re golden<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\">These equations model real-world change\u2014population growth, cooling coffee, stock trends. You&#8217;re learning to work backwards from &#8220;how it changes&#8221; to &#8220;what it actually is.&#8221;<\/p>\n<\/div>\n<section class=\"textbox interact\" aria-label=\"Interact\">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>.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558898\">Hint<\/button><\/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<\/section>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Order of Differential Equations<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Think of the &#8220;order&#8221; of a differential equation like ranking difficulty levels in a video game\u2014the higher the order, the more complex the equation becomes.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Simple Rule:<\/strong> The order equals the highest derivative that appears anywhere in the equation. That&#8217;s it!<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Quick Examples:<\/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]y' = 2x[\/latex] \u2192 <strong>First-order<\/strong> (highest derivative is [latex]y'[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y'' - 3y' + 2y = 0[\/latex] \u2192 <strong>Second-order<\/strong> (highest derivative is [latex]y''[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y''' + xy' = \\sin(x)[\/latex] \u2192 <strong>Third-order<\/strong> (highest derivative is [latex]y'''[\/latex])<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Common Notation to Watch For:<\/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]y'[\/latex], [latex]y''[\/latex], [latex]y'''[\/latex] (prime notation)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y^{(4)}[\/latex], [latex]y^{(5)}[\/latex] (parentheses for higher orders)<\/li>\n<li class=\"whitespace-normal break-words\">Sometimes written as [latex]\\frac{dy}{dx}[\/latex], [latex]\\frac{d^2y}{dx^2}[\/latex], etc.<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Why Order Matters:<\/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>First-order:<\/strong> Usually the most straightforward to solve<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Second-order:<\/strong> Common in physics (think springs, pendulums)<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Higher-order:<\/strong> More complex, often requiring specialized techniques<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Don&#8217;t get distracted by complicated-looking coefficients or messy right-hand sides. Just scan the equation for the highest derivative and you&#8217;ve got your answer.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\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\"><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-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\"><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-id1170571141543\" data-type=\"solution\">\n<p id=\"fs-id1170570993950\">[latex]5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">General Solutions vs. Particular Solutions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Think of general solutions like a recipe that says &#8220;add some salt to taste&#8221;\u2014there&#8217;s flexibility built in. A particular solution is like following that recipe and adding exactly 2 teaspoons of salt.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Key Distinction:<\/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>General Solution:<\/strong> Contains arbitrary constants (like [latex]C[\/latex]) and represents an entire family of functions<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Particular Solution:<\/strong> Has specific values for all constants\u2014just one function from the family<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Why does this happen? When you take derivatives, constants disappear. So when you work backwards from a derivative to find the original function, you need to account for any constant that could have been there originally.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Example:<\/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\">For [latex]y' = 2x[\/latex]:\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\">General: [latex]y = x^2 + C[\/latex] (infinite solutions)<\/li>\n<li class=\"whitespace-normal break-words\">Particular: [latex]y = x^2 + 3[\/latex] (one specific solution when [latex]C = 3[\/latex])<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\"><strong>Finding Particular Solutions:<\/strong><\/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\">Start with the general solution<\/li>\n<li class=\"whitespace-normal break-words\">Use given information (like a point the curve passes through)<\/li>\n<li class=\"whitespace-normal break-words\">Substitute and solve for the constant(s)<\/li>\n<li class=\"whitespace-normal break-words\">Replace [latex]C[\/latex] with the specific value<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558892\">Hint<\/button><\/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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558893\">Show Solution<\/button><\/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<\/section>\n<h2>Initial-Value Problems<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">What&#8217;s an Initial-Value Problem? It&#8217;s a differential equation plus one or more <strong>initial conditions<\/strong> that tell you specific values at a particular point (usually when [latex]t = 0[\/latex]).<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Magic Number Rule:<\/strong> You need exactly as many initial conditions as the order of your differential equation:<\/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>First-order equation<\/strong> \u2192 Need 1 initial condition<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Second-order equation<\/strong> \u2192 Need 2 initial conditions<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Third-order equation<\/strong> \u2192 Need 3 initial conditions<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">Why &#8220;Initial&#8221; Values? The independent variable often represents time, so [latex]t = 0[\/latex] is your starting point\u2014like knowing where a ball starts before tracking its motion.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Problem-Solving Strategy:<\/strong><\/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\"><strong>Solve the differential equation<\/strong> \u2192 Get the general solution (with constants like [latex]C[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Apply the initial condition(s)<\/strong> \u2192 Substitute given values to find the specific constants<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\">Your solution must satisfy both the differential equation AND the initial condition. Test both separately!<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1170573365823\" data-type=\"problem\">\n<p id=\"fs-id1170573429527\">Verify that [latex]y=3{e}^{2t}+4\\sin{t}[\/latex] is a solution to the initial-value problem<\/p>\n<div id=\"fs-id1170573368474\" class=\"unnumbered\" style=\"text-align: left;\" data-type=\"equation\" data-label=\"\">[latex]{y}^{\\prime }-2y=4\\cos{t} - 8\\sin{t},y\\left(0\\right)=3[\/latex].<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558890\">Hint<\/button><\/p>\n<div id=\"q44558890\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170571276500\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1170573338054\">First verify that [latex]y[\/latex] solves the differential equation. Then check the initial value.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1170573362845\" data-type=\"problem\">\n<p id=\"fs-id1170573362847\">Solve the initial-value problem<\/p>\n<div id=\"fs-id1170573570394\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]{y}^{\\prime }={x}^{2}-4x+3 - 6{e}^{x},y\\left(0\\right)=8[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558869\">Hint<\/button><\/p>\n<div id=\"q44558869\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170571418500\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1170573502068\">First take the antiderivative of both sides of the differential equation. Then substitute [latex]x=0[\/latex] and [latex]y=8[\/latex] into the resulting equation and solve for [latex]C[\/latex].<\/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=\"q44558879\">Show Solution<\/button><\/p>\n<div id=\"q44558879\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1170571455482\" data-type=\"solution\">\n<p id=\"fs-id1170571203542\">[latex]y=\\frac{1}{3}{x}^{3}-2{x}^{2}+3x - 6{e}^{x}+14[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":11,"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\/793"}],"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":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/793\/revisions"}],"predecessor-version":[{"id":2245,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/793\/revisions\/2245"}],"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\/793\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=793"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=793"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=793"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=793"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}