{"id":826,"date":"2025-10-09T15:41:54","date_gmt":"2025-10-09T15:41:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=826"},"modified":"2025-11-13T21:00:14","modified_gmt":"2025-11-13T21:00:14","slug":"nonhomogeneous-linear-equations-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/nonhomogeneous-linear-equations-fresh-take\/","title":{"raw":"Nonhomogeneous Linear Equations: Fresh Take","rendered":"Nonhomogeneous Linear Equations: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find general solutions of nonhomogeneous equations<\/li>\r\n \t<li>Apply the method of undetermined coefficients<\/li>\r\n \t<li>Apply the method of variation of parameters<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">General Solution to a Nonhomogeneous Linear Equation<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<div data-test-render-count=\"1\">\r\n<div>\r\n<div class=\"group relative pb-3\" data-is-streaming=\"false\">\r\n<div class=\"font-claude-response relative leading-[1.65rem] [&amp;_pre&gt;div]:bg-bg-000\/50 [&amp;_pre&gt;div]:border-0.5 [&amp;_pre&gt;div]:border-border-400 [&amp;_.ignore-pre-bg&gt;div]:bg-transparent [&amp;_.standard-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.standard-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8 [&amp;_.progressive-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.progressive-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8\">\r\n<div>\r\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 standard-markdown\">\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">A nonhomogeneous differential equation has the form [latex]a_2(x)y'' + a_1(x)y' + a_0(x)y = r(x)[\/latex], where [latex]r(x) \\neq 0[\/latex]. Unlike homogeneous equations, the right side is not zero, which changes how we find solutions.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Two Key Components<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">To solve a nonhomogeneous equation, you need two parts:<\/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-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>The complementary solution<\/strong> [latex]c_1y_1(x) + c_2y_2(x)[\/latex]: This is the general solution to the associated homogeneous equation (called the <strong>complementary equation<\/strong>), which you get by setting the right side equal to zero: [latex]a_2(x)y'' + a_1(x)y' + a_0(x)y = 0[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>A particular solution<\/strong> [latex]y_p(x)[\/latex]: This is any single solution to the nonhomogeneous equation itself. It contains no arbitrary constants.<\/li>\r\n<\/ol>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Building the General Solution<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Once you have both components, combine them to form the general solution:<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]y(x) = c_1y_1(x) + c_2y_2(x) + y_p(x)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The complementary solution handles the homogeneous part, while the particular solution accounts for the nonzero right side. Together, they capture all possible solutions to the nonhomogeneous equation.<\/p>\r\n<p class=\"font-claude-response-body 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-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the complementary equation by setting [latex]r(x) = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the complementary equation to find [latex]c_1y_1(x) + c_2y_2(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find one particular solution [latex]y_p(x)[\/latex] to the nonhomogeneous equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Combine them: [latex]y(x) = c_1y_1(x) + c_2y_2(x) + y_p(x)[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The challenge lies in step 3\u2014finding [latex]y_p(x)[\/latex]. Different methods exist for this, depending on the form of [latex]r(x)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Given that [latex]y_p(x)=-2[\/latex] is a particular solution to [latex]y^{\\prime\\prime}-3y^\\prime-4y=8[\/latex], write the general solution and verify that the general solution satisfies the equation.[reveal-answer q=\"039452983\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"039452983\"][latex]y(x)=c_1e^{-x}+c_2e^{4x}-2[\/latex][\/hidden-answer]<\/section>\r\n<h2 data-type=\"title\">Undetermined Coefficients<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">The method of undetermined coefficients finds a particular solution [latex]y_p(x)[\/latex] by making an educated guess based on the form of [latex]r(x)[\/latex]. The key insight is that derivatives of polynomials, exponentials, sines, and cosines produce the same types of functions. Your guess should match the form of [latex]r(x)[\/latex], with undetermined coefficients that you solve for by substitution.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Making Your Initial Guess<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">For polynomials, guess a polynomial of the same degree with all lower-order terms included (even if absent in [latex]r(x)[\/latex]). For [latex]r(x) = 3x[\/latex], guess [latex]y_p = Ax + B[\/latex], not just [latex]Ax[\/latex].<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">For exponentials like [latex]r(x) = ae^{\\lambda x}[\/latex], guess [latex]y_p = Ae^{\\lambda x}[\/latex].<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">For trigonometric functions, always include both sine and cosine terms in your guess, even if [latex]r(x)[\/latex] contains only one. For [latex]r(x) = a\\cos\\beta x + b\\sin\\beta x[\/latex], guess [latex]y_p = A\\cos\\beta x + B\\sin\\beta x[\/latex].<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">For products (like polynomials times exponentials or exponentials times trig functions), your guess should be a product with the same structure.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>The Critical Conflict Check<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Before finalizing your guess, check whether any term in [latex]y_p(x)[\/latex] solves the complementary equation. If there's a conflict, multiply your entire guess by [latex]x[\/latex]. If the conflict persists, multiply by [latex]x[\/latex] again. Repeat until no term in your guess solves the complementary equation.<\/p>\r\n<p class=\"font-claude-response-body 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-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Solve the complementary equation and write its general solution<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Make an initial guess for [latex]y_p(x)[\/latex] based on the form of [latex]r(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check for conflicts with the complementary solution; multiply by [latex]x[\/latex] if needed<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute [latex]y_p(x)[\/latex] into the differential equation and equate coefficients of like terms to solve for the unknown coefficients<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write the general solution: complementary solution plus particular solution<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the general solution to [latex]y^{\\prime\\prime}-4y^\\prime+4y=7\\sin t-\\cos t[\/latex].[reveal-answer q=\"028934924\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"028934924\"][latex]y(t)=c_1e^{2t}+c_2te^{2t}+\\sin t+\\cos t[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8250342&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=rQsUWvHvRSY&amp;video_target=tpm-plugin-hfu2pzjl-rQsUWvHvRSY\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP7.11_transcript.html\">transcript for \u201cCP 7.11\u201d here (opens in new window).<\/a><\/center><\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1170571715424\">Find the general solution to the following differential equations.<\/p>\r\n\r\n<ol id=\"fs-id1170571715427\" type=\"a\">\r\n \t<li>[latex]y^{\\prime\\prime}-5y^\\prime+4y=3e^x[\/latex]<\/li>\r\n \t<li>[latex]y^{\\prime\\prime}+y^\\prime-6y=52\\cos2t[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"597293813\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"597293813\"]\r\n\r\na.\u00a0[latex]y(x)=c_1e^{4x}+c_2e^x-xe^x[\/latex]\r\n\r\nb.\u00a0[latex]y(t)=c_1e^t+c_2te^t+te^t\\ln|t|[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Variation of Parameters<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">When [latex]r(x)[\/latex] is not a combination of polynomials, exponentials, sines, or cosines, the method of undetermined coefficients fails. The <strong>method of variation of parameters<\/strong> works for any continuous [latex]r(x)[\/latex], making it more versatile.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>The Core Idea<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Start with the general solution to the complementary equation: [latex]c_1y_1(x) + c_2y_2(x)[\/latex]. Instead of using constants [latex]c_1[\/latex] and [latex]c_2[\/latex], replace them with functions [latex]u(x)[\/latex] and [latex]v(x)[\/latex] to form the particular solution: [latex]y_p(x) = u(x)y_1(x) + v(x)y_2(x)[\/latex]. The goal is to find [latex]u(x)[\/latex] and [latex]v(x)[\/latex] such that [latex]y_p[\/latex] satisfies the nonhomogeneous equation.<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Finding u and v<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">First, ensure your differential equation has a leading coefficient of 1 (divide through by [latex]a[\/latex] if needed). The functions [latex]u(x)[\/latex] and [latex]v(x)[\/latex] come from solving this system:<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\begin{aligned} u'y_1 + v'y_2 &amp;= 0 \\ u'y_1' + v'y_2' &amp;= r(x) \\end{aligned}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">Use Cramer's rule to solve for [latex]u'[\/latex] and [latex]v'[\/latex], then integrate to find [latex]u[\/latex] and [latex]v[\/latex]. The particular solution is [latex]y_p(x) = u(x)y_1(x) + v(x)y_2(x)[\/latex].<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Cramer's Rule Reminder<\/strong><\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\">For the system [latex]\\begin{aligned} a_1z_1 + b_1z_2 &amp;= r_1 \\ a_2z_1 + b_2z_2 &amp;= r_2 \\end{aligned}[\/latex], the solutions are:<\/p>\r\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]z_1 = \\frac{\\begin{vmatrix}r_1 &amp; b_1 \\\\ r_2 &amp; b_2\\end{vmatrix}}{\\begin{vmatrix}a_1 &amp; b_1 \\\\ a_2 &amp; b_2\\end{vmatrix}}[\/latex] and [latex]z_2 = \\frac{\\begin{vmatrix}a_1 &amp; r_1 \\\\ a_2 &amp; r_2\\end{vmatrix}}{\\begin{vmatrix}a_1 &amp; b_1 \\\\ a_2 &amp; b_2\\end{vmatrix}}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body 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-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Solve the complementary equation to find [latex]y_1(x)[\/latex] and [latex]y_2(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set up and solve the system for [latex]u'[\/latex] and [latex]v'[\/latex] using Cramer's rule<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Integrate to find [latex]u(x)[\/latex] and [latex]v(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Form the particular solution: [latex]y_p(x) = u(x)y_1(x) + v(x)y_2(x)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write the general solution: [latex]y(x) = c_1y_1(x) + c_2y_2(x) + y_p(x)[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1170571553660\">Use Cramer\u2019s rule to solve the following system of equations.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}\r\n2xz_1-3z_2&amp;=0 \\\\\r\nx^2z_1+4xz_2&amp;=x+1\r\n\\end{aligned}[\/latex].<\/p>\r\n[reveal-answer q=\"734571894\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"734571894\"]\r\n<p style=\"text-align: center;\">[latex]z_1=\\frac{3x+3}{11x^2}, \\ z_2\\frac{2x+2}{11x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1170572554234\">Find the general solution to the following differential equations.<\/p>\r\n\r\n<ol id=\"fs-id1170572554237\" type=\"a\">\r\n \t<li>[latex]y^{\\prime\\prime}+y=\\sec x[\/latex]<\/li>\r\n \t<li>[latex]x^{\\prime\\prime}-2x^\\prime+x=\\frac{e^t}t[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"687247112\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"687247112\"]\r\n\r\na.\u00a0[latex]y(x)=c_1\\cos x+c_2\\sin x+\\cos x\\ln|\\cos x|+x\\sin x[\/latex]\r\n\r\nb.\u00a0[latex]x(t)=c_1e^t+c_2te^t+te^t\\ln|t|[\/latex]\r\n\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 src=\"\/\/plugin.3playmedia.com\/show?mf=8250341&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=8N2A8lhcxPI&amp;video_target=tpm-plugin-y9gjmauh-8N2A8lhcxPI\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP7.14a_transcript.html\">transcript for \u201cCP 7.14a\u201d here (opens in new window).<\/a><\/center><center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8250340&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=HCMJodmpYKg&amp;video_target=tpm-plugin-4khktbip-HCMJodmpYKg\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP7.14b_transcript.html\">transcript for \u201cCP 7.14b\u201d here (opens in new window).<\/a><\/center><\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find general solutions of nonhomogeneous equations<\/li>\n<li>Apply the method of undetermined coefficients<\/li>\n<li>Apply the method of variation of parameters<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">General Solution to a Nonhomogeneous Linear Equation<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<div data-test-render-count=\"1\">\n<div>\n<div class=\"group relative pb-3\" data-is-streaming=\"false\">\n<div class=\"font-claude-response relative leading-[1.65rem] [&amp;_pre&gt;div]:bg-bg-000\/50 [&amp;_pre&gt;div]:border-0.5 [&amp;_pre&gt;div]:border-border-400 [&amp;_.ignore-pre-bg&gt;div]:bg-transparent [&amp;_.standard-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.standard-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8 [&amp;_.progressive-markdown_:is(p,blockquote,h1,h2,h3,h4,h5,h6)]:pl-2 [&amp;_.progressive-markdown_:is(p,blockquote,ul,ol,h1,h2,h3,h4,h5,h6)]:pr-8\">\n<div>\n<div class=\"grid-cols-1 grid gap-2.5 [&amp;_&gt;_*]:min-w-0 standard-markdown\">\n<p class=\"font-claude-response-body whitespace-normal break-words\">A nonhomogeneous differential equation has the form [latex]a_2(x)y'' + a_1(x)y' + a_0(x)y = r(x)[\/latex], where [latex]r(x) \\neq 0[\/latex]. Unlike homogeneous equations, the right side is not zero, which changes how we find solutions.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Two Key Components<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">To solve a nonhomogeneous equation, you need two parts:<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>The complementary solution<\/strong> [latex]c_1y_1(x) + c_2y_2(x)[\/latex]: This is the general solution to the associated homogeneous equation (called the <strong>complementary equation<\/strong>), which you get by setting the right side equal to zero: [latex]a_2(x)y'' + a_1(x)y' + a_0(x)y = 0[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\"><strong>A particular solution<\/strong> [latex]y_p(x)[\/latex]: This is any single solution to the nonhomogeneous equation itself. It contains no arbitrary constants.<\/li>\n<\/ol>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Building the General Solution<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Once you have both components, combine them to form the general solution:<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]y(x) = c_1y_1(x) + c_2y_2(x) + y_p(x)[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The complementary solution handles the homogeneous part, while the particular solution accounts for the nonzero right side. Together, they capture all possible solutions to the nonhomogeneous equation.<\/p>\n<p class=\"font-claude-response-body 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-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Identify the complementary equation by setting [latex]r(x) = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Solve the complementary equation to find [latex]c_1y_1(x) + c_2y_2(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Find one particular solution [latex]y_p(x)[\/latex] to the nonhomogeneous equation<\/li>\n<li class=\"whitespace-normal break-words\">Combine them: [latex]y(x) = c_1y_1(x) + c_2y_2(x) + y_p(x)[\/latex]<\/li>\n<\/ol>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The challenge lies in step 3\u2014finding [latex]y_p(x)[\/latex]. Different methods exist for this, depending on the form of [latex]r(x)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Given that [latex]y_p(x)=-2[\/latex] is a particular solution to [latex]y^{\\prime\\prime}-3y^\\prime-4y=8[\/latex], write the general solution and verify that the general solution satisfies the equation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q039452983\">Show Solution<\/button><\/p>\n<div id=\"q039452983\" class=\"hidden-answer\" style=\"display: none\">[latex]y(x)=c_1e^{-x}+c_2e^{4x}-2[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2 data-type=\"title\">Undetermined Coefficients<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">The method of undetermined coefficients finds a particular solution [latex]y_p(x)[\/latex] by making an educated guess based on the form of [latex]r(x)[\/latex]. The key insight is that derivatives of polynomials, exponentials, sines, and cosines produce the same types of functions. Your guess should match the form of [latex]r(x)[\/latex], with undetermined coefficients that you solve for by substitution.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Making Your Initial Guess<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">For polynomials, guess a polynomial of the same degree with all lower-order terms included (even if absent in [latex]r(x)[\/latex]). For [latex]r(x) = 3x[\/latex], guess [latex]y_p = Ax + B[\/latex], not just [latex]Ax[\/latex].<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">For exponentials like [latex]r(x) = ae^{\\lambda x}[\/latex], guess [latex]y_p = Ae^{\\lambda x}[\/latex].<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">For trigonometric functions, always include both sine and cosine terms in your guess, even if [latex]r(x)[\/latex] contains only one. For [latex]r(x) = a\\cos\\beta x + b\\sin\\beta x[\/latex], guess [latex]y_p = A\\cos\\beta x + B\\sin\\beta x[\/latex].<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">For products (like polynomials times exponentials or exponentials times trig functions), your guess should be a product with the same structure.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>The Critical Conflict Check<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Before finalizing your guess, check whether any term in [latex]y_p(x)[\/latex] solves the complementary equation. If there&#8217;s a conflict, multiply your entire guess by [latex]x[\/latex]. If the conflict persists, multiply by [latex]x[\/latex] again. Repeat until no term in your guess solves the complementary equation.<\/p>\n<p class=\"font-claude-response-body 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-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Solve the complementary equation and write its general solution<\/li>\n<li class=\"whitespace-normal break-words\">Make an initial guess for [latex]y_p(x)[\/latex] based on the form of [latex]r(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Check for conflicts with the complementary solution; multiply by [latex]x[\/latex] if needed<\/li>\n<li class=\"whitespace-normal break-words\">Substitute [latex]y_p(x)[\/latex] into the differential equation and equate coefficients of like terms to solve for the unknown coefficients<\/li>\n<li class=\"whitespace-normal break-words\">Write the general solution: complementary solution plus particular solution<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the general solution to [latex]y^{\\prime\\prime}-4y^\\prime+4y=7\\sin t-\\cos t[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q028934924\">Show Solution<\/button><\/p>\n<div id=\"q028934924\" class=\"hidden-answer\" style=\"display: none\">[latex]y(t)=c_1e^{2t}+c_2te^{2t}+\\sin t+\\cos t[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the above example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8250342&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=rQsUWvHvRSY&amp;video_target=tpm-plugin-hfu2pzjl-rQsUWvHvRSY\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP7.11_transcript.html\">transcript for \u201cCP 7.11\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1170571715424\">Find the general solution to the following differential equations.<\/p>\n<ol id=\"fs-id1170571715427\" type=\"a\">\n<li>[latex]y^{\\prime\\prime}-5y^\\prime+4y=3e^x[\/latex]<\/li>\n<li>[latex]y^{\\prime\\prime}+y^\\prime-6y=52\\cos2t[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q597293813\">Show Solution<\/button><\/p>\n<div id=\"q597293813\" class=\"hidden-answer\" style=\"display: none\">\n<p>a.\u00a0[latex]y(x)=c_1e^{4x}+c_2e^x-xe^x[\/latex]<\/p>\n<p>b.\u00a0[latex]y(t)=c_1e^t+c_2te^t+te^t\\ln|t|[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2 data-type=\"title\">Variation of Parameters<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">When [latex]r(x)[\/latex] is not a combination of polynomials, exponentials, sines, or cosines, the method of undetermined coefficients fails. The <strong>method of variation of parameters<\/strong> works for any continuous [latex]r(x)[\/latex], making it more versatile.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>The Core Idea<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Start with the general solution to the complementary equation: [latex]c_1y_1(x) + c_2y_2(x)[\/latex]. Instead of using constants [latex]c_1[\/latex] and [latex]c_2[\/latex], replace them with functions [latex]u(x)[\/latex] and [latex]v(x)[\/latex] to form the particular solution: [latex]y_p(x) = u(x)y_1(x) + v(x)y_2(x)[\/latex]. The goal is to find [latex]u(x)[\/latex] and [latex]v(x)[\/latex] such that [latex]y_p[\/latex] satisfies the nonhomogeneous equation.<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Finding u and v<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">First, ensure your differential equation has a leading coefficient of 1 (divide through by [latex]a[\/latex] if needed). The functions [latex]u(x)[\/latex] and [latex]v(x)[\/latex] come from solving this system:<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]\\begin{aligned} u'y_1 + v'y_2 &= 0 \\ u'y_1' + v'y_2' &= r(x) \\end{aligned}[\/latex]<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">Use Cramer&#8217;s rule to solve for [latex]u'[\/latex] and [latex]v'[\/latex], then integrate to find [latex]u[\/latex] and [latex]v[\/latex]. The particular solution is [latex]y_p(x) = u(x)y_1(x) + v(x)y_2(x)[\/latex].<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\"><strong>Cramer&#8217;s Rule Reminder<\/strong><\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\">For the system [latex]\\begin{aligned} a_1z_1 + b_1z_2 &= r_1 \\ a_2z_1 + b_2z_2 &= r_2 \\end{aligned}[\/latex], the solutions are:<\/p>\n<p class=\"font-claude-response-body whitespace-normal break-words\" style=\"text-align: center;\">[latex]z_1 = \\frac{\\begin{vmatrix}r_1 & b_1 \\\\ r_2 & b_2\\end{vmatrix}}{\\begin{vmatrix}a_1 & b_1 \\\\ a_2 & b_2\\end{vmatrix}}[\/latex] and [latex]z_2 = \\frac{\\begin{vmatrix}a_1 & r_1 \\\\ a_2 & r_2\\end{vmatrix}}{\\begin{vmatrix}a_1 & b_1 \\\\ a_2 & b_2\\end{vmatrix}}[\/latex]<\/p>\n<p class=\"font-claude-response-body 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-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Solve the complementary equation to find [latex]y_1(x)[\/latex] and [latex]y_2(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Set up and solve the system for [latex]u'[\/latex] and [latex]v'[\/latex] using Cramer&#8217;s rule<\/li>\n<li class=\"whitespace-normal break-words\">Integrate to find [latex]u(x)[\/latex] and [latex]v(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Form the particular solution: [latex]y_p(x) = u(x)y_1(x) + v(x)y_2(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Write the general solution: [latex]y(x) = c_1y_1(x) + c_2y_2(x) + y_p(x)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1170571553660\">Use Cramer\u2019s rule to solve the following system of equations.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}  2xz_1-3z_2&=0 \\\\  x^2z_1+4xz_2&=x+1  \\end{aligned}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q734571894\">Show Solution<\/button><\/p>\n<div id=\"q734571894\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]z_1=\\frac{3x+3}{11x^2}, \\ z_2\\frac{2x+2}{11x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1170572554234\">Find the general solution to the following differential equations.<\/p>\n<ol id=\"fs-id1170572554237\" type=\"a\">\n<li>[latex]y^{\\prime\\prime}+y=\\sec x[\/latex]<\/li>\n<li>[latex]x^{\\prime\\prime}-2x^\\prime+x=\\frac{e^t}t[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q687247112\">Show Solution<\/button><\/p>\n<div id=\"q687247112\" class=\"hidden-answer\" style=\"display: none\">\n<p>a.\u00a0[latex]y(x)=c_1\\cos x+c_2\\sin x+\\cos x\\ln|\\cos x|+x\\sin x[\/latex]<\/p>\n<p>b.\u00a0[latex]x(t)=c_1e^t+c_2te^t+te^t\\ln|t|[\/latex]<\/p>\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\" src=\"\/\/plugin.3playmedia.com\/show?mf=8250341&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=8N2A8lhcxPI&amp;video_target=tpm-plugin-y9gjmauh-8N2A8lhcxPI\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP7.14a_transcript.html\">transcript for \u201cCP 7.14a\u201d here (opens in new window).<\/a><\/div>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8250340&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=HCMJodmpYKg&amp;video_target=tpm-plugin-4khktbip-HCMJodmpYKg\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP7.14b_transcript.html\">transcript for \u201cCP 7.14b\u201d here (opens in new window).<\/a><\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":16,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-826","chapter","type-chapter","status-publish","hentry"],"part":440,"module-header":"fresh_take","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\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/826","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/826\/revisions"}],"predecessor-version":[{"id":2266,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/826\/revisions\/2266"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/440"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/826\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=826"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=826"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=826"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=826"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}