{"id":233,"date":"2019-08-15T17:03:07","date_gmt":"2019-08-15T17:03:07","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=233"},"modified":"2020-06-18T17:19:18","modified_gmt":"2020-06-18T17:19:18","slug":"first-order-differential-equations","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/differential-equations\/first-order-differential-equations\/","title":{"rendered":"First-Order Differential Equations"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<p>Consider the first-order ODE, which is when the highest derivative appearing in the equation is a first derivative.<\/p>\n\n\n\n<center><p>  $y&#8217;=f(t,y)$, $y(t_{0})=y_{0}$ <\/p><\/center>\n\n\n\n<ol class=\"wp-block-list\"><li> Does $y&#8217;=f(t,y)$, $y(t_{0})=y_{0}$ have a solution?\n\n\t<br><br>\n\t<\/li><li> If so, can we find a formula for the solution?\n\n<\/li><\/ol>\n\n\n\n<p>\nThe first question is easily addressed:\n\n<\/p>\n\n\n\n<center>\n<p align=\"center\">\n\t\t<\/p><center><h4><b>Existence and Uniqueness Theorem<\/b><\/h4><\/center>\n\n\t\t<p>\n\t\tSuppose that $f(t,y)$ and $\\frac{\\partial f(t,y)}{\\partial y}$ are\n\t\tcontinumous on a closed rectangle $R$ of the $ty$-plane.  If\n\t\t$(t_{0},y_{0}) \\in R$, then the <b>IVP<\/b>, an ODE along with a specified initial condition,\n\t\t$$\n\t\ty&#8217;=f(t,y), \\quad y(t_{0})=y_{0}\n\t\t$$\n\t\thas a unique solution $y(t)$ on some $t$-interval containing $t_{0}$.\n<\/p>\n<\/center>\n\n\n\n<p>\nThe second question is much more difficult, and often we need to\nresort to numerical methods.  However, in this tutorial we review four\nof the most commonly-used analytic solution methods for first-order\nODES.\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Separating the Variables<\/h4>\n\n\n\n<p>If an ODE can be written in the form $$ \\frac{\\partial y}{\\partial t}=\\frac{g(t)}{h(y)}, $$ then the ODE is said to be <b>separable<\/b>.  In this case, a simple solution technique can be derived as follows:  <\/p>\n\n\n\n<p>\nSuppose $y=f(t)$ solves the ODE.  Rewriting the ODE as \n$h(y)y&#8217;=g(t)$, \n\n<\/p>\n\n\n\n<p> $$\\begin{array}{rcl} h(f(t))f'(t) &amp;= &amp;g(t) \\\\ \\int h(f(t))f'(t)dt &amp;= &amp;\\int g(t)dt +C \\\\ \\int h(y)dy &amp; = &amp; \\int g(t)dt +C.  \\end{array}$$ <\/p>\n\n\n\n<p>\n\t\t\t\tSince $y=f(t)$, $y&#8217;=f'(t)$.\n<\/p>\n\n\n\n<p>\n\t\t\t\tIntegrating with respect to $t$ on each side.\n<\/p>\n\n\n\n<p> Since $dy=f'(t)dt$. <\/p>\n\n\n\n<p>Upon integrating, we have our&nbsp;<strong>implicitly-defined<\/strong>&nbsp;general solution of the ODE, which we can often solve explicitly for&nbsp;<em>y<\/em>(<em>t<\/em>).  A solution of an ODE is said to be written implicitly if it has the form $H(xy)=C$, rather than being solved for y in terms of x.<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nLet&#8217;s solve the separable ODE $y&#8217; = \\frac{4y}{t}$.  Separating the\nvariables and integrating, \n\n<\/p>\n\n\n\n<p> $$\\begin{array}{rcl} \\int \\frac{dy}{4y} &amp;= &amp;\\int \\frac{dt}{t} +C_{1}, \\\\ \\frac{1}{4} \\ln |y| &amp;= &amp;\\ln |t| +C_{1}, \\\\ \\frac{1}{4} \\ln |y| &#8211;  \\ln |t| &amp;= &amp;C_{1} \\\\ \\ln \\left| \\frac{y^{1\/4}}{t} \\right| &amp; = &amp; C_{1} \\\\ e^{\\ln \\left| \\frac{y^{1\/4}}{t} \\right|} &amp; = &amp; e^{C_{1}} \\\\ \\left| \\frac{y^{1\/4}}{t} \\right| &amp; = &amp; C_{2} \\\\ y^{1\/4} &amp; = &amp; C_{2}t &amp; \\\\ y &amp; = &amp; Ct^{4}.  \\end{array}$$<\/p>\n\n\n\n<p>\n\t\t\t\tRecall that <br> $r\\ln a = \\ln a^{r} \\mathrm{~and~}$ <br> $\\ln a &#8211; \\ln b = \\ln \\frac{a}{b}.$\n<\/p>\n\n\n\n<p>\n\t\t\t\t$e^{\\ln a}= a$. <br> Relabel $e^{C_{1}}$ as $C_{2}$.\n<\/p>\n\n\n\n<p>\n\t\t\t\tRelabel $C_{2}^{4}$ as $C$.\n<\/p>\n\n\n\n<p>\nThe general solution, $y=Ct^4$, defines a family of solution cuves\ncorresponding to various initial conditions.\n\n<\/p>\n\n\n<style>.kt-accordion-id_abdf53-9b .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_abdf53-9b .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_abdf53-9b > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_abdf53-9b:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_abdf53-9b:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap 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.kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_abdf53-9b:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_abdf53-9b:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_abdf53-9b .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_abdf53-9b .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_abdf53-9b kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_2c1b63-4f\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">View Solution<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/odetut1.gif?resize=549%2C404&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-376\" width=\"549\" height=\"404\"\/><\/figure><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Using an Integrating Factor to solve a Linear ODE<\/h4>\n\n\n\n<p>If a first-order ODE can be written in the <b>normal linear form<\/b> $$ y&#8217;+p(t)y= q(t), $$ the ODE can be solved using an integrating factor $\\mu (t)= e^{\\int p(t)dt}$: <\/p>\n\n\n\n<p>\n\t\t\t\tMultiplying both sides of the ODE by $\\mu (t)$.\n<\/p>\n\n\n\n<p> $\\left( \\mu (t)y \\right) &#8216; = \\mu (t)y&#8217; + \\mu &#8216;(t)y$ and $\\mu &#8216;(t) = p(t) \\mu (t)$ using the chain rule to differentiate $\\mu (t)= e^{\\int p(t)dt}$. <\/p>\n\n\n\n<p>\n\t\t\t\tIntegrating each side with respect to $t$.\n<\/p>\n\n\n\n<p>\nDividing through by $\\mu (t)$, we have the general solution of the\nlinear ODE.\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Example<\/h4>\n\n\n\n<p>\n\nWe can solve the linear ODE $y&#8217;-2ty=t$ using an integrating facter.\nHere, $p(t)=-2t$ and $q(t)=1$, so\n$$\n\\mu (t) = e^{\\int -2t dt}=e^{-t^{2}}\n$$\nMultiplying both sides of the ODE by $\\mu (t)$,\n\n<\/p>\n\n\n\n<p> $\\begin{array}{rcl} e^{-t^{2}}(y&#8217;-2ty) &amp; = &amp; te^{-t^{2}} \\\\ (e^{-t^{2}}y)&#8217; &amp; = &amp;  te^{-t^{2}} \\\\ e^{-t^{2}}y &amp; = &amp; -\\frac{1}{2}e^{-t^{2}}+C \\\\ y &amp; = &amp; Ce^{t^{2}}-\\frac{1}{2}. \\end{array}$ <\/p>\n\n\n\n<p>\n\t\t\t\tYou can verify that $(e^{-t^{2}}y)&#8217;=e^{-t^{2}}(y&#8217;-2ty)$\n\t\t\t\tby using the Chain Rule to differentiate $e^{-t^{2}}y$\n<\/p>\n\n\n\n<p>\n\t\t\t\tIntegrating each side with respect to $t$.\n<\/p>\n\n\n\n<p>\n\t\t\t\tDividing through by $e^{-t^{2}}$.\n<\/p>\n\n\n\n<p>\nSo the general solution of $y&#8217;-2ty=t$ is $y(t)=Ce^{t^{2}}-\\frac{1}{2}$.\n  \n<\/p>\n\n\n\n<p>\nFor practice, solve $y&#8217; = \\frac{4y}{t}$ by putting it in normal linear\nform and using an integrating factor.  Verify that you get the same\nresult as we did by separating the variables.\n\n<\/p>\n\n\n<style>.kt-accordion-id_fb9351-ec .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_fb9351-ec .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_fb9351-ec > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > 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.kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_fb9351-ec:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_fb9351-ec:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_fb9351-ec:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_fb9351-ec:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_fb9351-ec:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_fb9351-ec:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_fb9351-ec:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_fb9351-ec:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_fb9351-ec:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_fb9351-ec:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_fb9351-ec .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_fb9351-ec > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_fb9351-ec:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_fb9351-ec:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_fb9351-ec:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_fb9351-ec:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_fb9351-ec:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_fb9351-ec .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_fb9351-ec .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_fb9351-ec kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_9bd5e6-11\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">View Solution<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/odetut2.gif?resize=502%2C380&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-377\" width=\"502\" height=\"380\"\/><\/figure><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Using a Change of Variables<\/h4>\n\n\n\n<p>\n\nOften, a first-order ODE that is neither separable nor linear can be\nsimplified to one of these types by making a change of variables.\nHere are some important examples:\n\n<\/p>\n\n\n\n<p><b>Homogeneous Equation of Order 0<\/b>:  $\\frac{dy}{dx} = f(x,y)$ where $f(kx,ky)= f(x,y)$.  Use the change of variables $z=\\frac{y}{x}$ to convert the ODE to $x\\frac{dz}{dx} = f(1,z)-z$, which is separable.<\/p>\n\n\n<style>.kt-accordion-id_a2ab53-91 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_a2ab53-91 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_a2ab53-91 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_a2ab53-91 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_a2ab53-91 .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_a2ab53-91 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_a2ab53-91 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_a2ab53-91:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_a2ab53-91 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_a2ab53-91 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_a2ab53-91 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_9c5a68-c2\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Derivation<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>Let $z=y\/x$.  Then $y=xz$, and<\/p>\n\n\n\n<p>$$\\frac{dy}{dx}=x\\frac{dz}{dx}+z.$$ By the Chain Rule.<\/p>\n\n\n\n<p>Substituting into the original equation,<br><\/p>\n\n\n\n<p>\\begin{eqnarray*}<br>x\\frac{dz}{dx}+z &amp; =&amp; f(x,xz) \\<br>         x\\frac{dz}{dx}+z  &amp;= &amp;  f(1,z)\\<br>         x\\frac{dz}{dx} &amp; = &amp; f(1,z) -z.<br> \\end{eqnarray*}<\/p>\n\n\n\n<p><br>Since $f(kx,ky)=f(x,y)$ for all $k$.<\/p>\n\n\n\n<p>We recommend understanding the change of variables procedure, rather<br> than memorizing the equation that results! After solving this separable equation, use $z=y\/x$ to return to the original variables. <\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p><br><b>Bernoulli Equation<\/b>:  $\\frac{dy}{dt}+p(t)y= q(t)y^{b} \\quad<br>(b \\not= 0,1)$. Use the change of variables $z=y^{1-b}$ to convert the ODE to $\\frac{dz}{dt} + (1-b)p(t)z=(1-b)q(t)$, which is linear.<\/p>\n\n\n<style>.kt-accordion-id_88fcf7-5a .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_88fcf7-5a .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_88fcf7-5a > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_88fcf7-5a > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_88fcf7-5a .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_88fcf7-5a .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_88fcf7-5a > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_88fcf7-5a:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_88fcf7-5a .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_88fcf7-5a .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_88fcf7-5a kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_b23485-e4\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Derivation<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>Let $z=y^{1-b}$.  Then $y=z^{\\frac{1}{1-b}}$, so by the Chain Rule,<br>\n$$<br>\n\\frac{dy}{dt}= \\frac{1}{1-b}z^{\\frac{1}{1-b}-1}\\frac{dz}{dt}=\\frac{1}{1-b}z^{\\frac{b}{1-b}}\\frac{dz}{dt}.<br>\n$$<br>\nSubstituting into the original equation,<\/p>\n\n\n\n<p>\\begin{eqnarray}         \\frac{1}{1-b} z^{\\frac{b}{1-b}}\\frac{dz}{dt} +          p(t)z^{\\frac{1}{1-b}} &amp;         =  &amp; q(t)z^{\\frac{b}{1-b}} \\         \\frac{1}{1-b}\\frac{dz}{dt} + p(t)z &amp; = &amp; q(t)\\         \\frac{dz}{dt} + (1-b)p(t)z &amp; = &amp; (1-b)q(t) .     \\end{eqnarray}<br>Dividing through by $z^{\\frac{b}{1-b}}.<\/p>\n\n\n\n<p>After solving this linear equation, use $z= y^{1-b}$ to return to the<br>\noriginal variables.<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p><br><b>Riccati Equation<\/b>:  $\\frac{dy}{dt}=a(t)y+b(t)y^{2}+ F(t)$. If one particular solution $g(t)$ is known, use the change of variables $z=\\frac{1}{y-g}$ to convert the ODE to $\\frac{dz}{dt}+ (a+2bg)z=-b$, which is linear.<\/p>\n\n\n<style>.kt-accordion-id_865cca-79 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_865cca-79 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_865cca-79 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_865cca-79 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_865cca-79 .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_865cca-79 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_865cca-79 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_865cca-79:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_865cca-79 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_865cca-79 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_865cca-79 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_688f6e-21\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Derivation<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>Suppose that $y=g(t)$ solves the ODE.<\/p>\n\n\n\n<p>Let $z=\\frac{1}{y-g}$.  Then $y=g(t) + \\frac{1}{z}$, and<br>\n$$<br>\n\\frac{dy}{dt}=g'(t) -\\frac{1}{z^{2}}\\frac{dz}{dt}.<br>\n$$<\/p>\n\n\n\n<p>Substituting into the original ODE,<\/p>\n\n\n\n<p>\\begin{eqnarray}         g'(t) -\\frac{1}{z^{2}}\\frac{dz}{dt} &amp; = &amp; a(t)\\left[ g(t)+\\frac{1}{z}          \\right] + b(t)\\left[ g(t)+\\frac{1}{z} \\right] ^{2} + F(t)\\         g'(t) -\\frac{1}{z^{2}}\\frac{dz}{dt} &amp; = &amp; a(t)g(t) + a(t)\\frac{1}{z} +          b(t)\\left( g(t) \\right) ^{2} + 2b(t)g(t)\\frac{1}{z} +          b(t)\\frac{1}{z^{2}}+ F(t)  \\          -\\frac{1}{z^{2}}\\frac{dz}{dt} &amp; = &amp; \\left[ a(t)g(t) + b(t)\\left( g(t)          \\right) ^{2} + F(t) -g'(t) \\right] + \\left( a(t) + 2b(t)g(t) \\right)          \\frac{1}{z} + b(t)\\frac{1}{z^{2}}  \\         -\\frac{1}{z^{2}}\\frac{dz}{dt} &amp; = &amp; (a+2bg)\\frac{1}{z} +          \\frac{b}{z^{2}} \\         \\frac{dz}{dt} &amp; = &amp; -(a+2bg)z &#8211; b \\         \\frac{dz}{dt} + (a+2bg)z &amp; = &amp; -b .     \\end{eqnarray}<\/p>\n\n\n\n<p>$g'(t) = ag+ bg^{2} + F(t)$, since $g(t)$ solves<br>\n     the original ODE.<\/p>\n\n\n\n<p>After solving this linear equation, use $y=g(t) + \\frac{1}{z}$ to<br>\nreturn to the original variables.<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p><br>When using a change of variables, solve the transformed ODE and then return to the original variables to obtain the general solution of the original ODE.  Often, you will have to leave your solution in implicit form.  <\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Example<\/h4>\n\n\n\n<p>\n\nLet&#8217;s solve the ODE $\\frac{dy}{dx}=\\frac{y-x}{x-4y}$.  To see that it\nis homogeneous of order 0, note that\n$$\nf(kx,ky)= \\frac{ky-kx}{kx-4ky}=\\frac{y-x}{x-4y}=f(x,y).\n$$\n\n<\/p>\n\n\n\n<p>\nLet $z=\\frac{y}{x}$.  Then $y=xz$, so\n$\\frac{dy}{dx}=x\\frac{dz}{dx}+z$.  The ODE becomes\n\\begin{eqnarray*}\n\tx\\frac{dz}{dx}+z &amp;= &amp; \\frac{xz-x}{x-4xz} \\\\\n\tx\\frac{dz}{dx}+z &amp;= &amp; \\frac{z-1}{1-4z}\\\\\n\tx\\frac{dz}{dx} &amp; = &amp; \\frac{4z^{2}-1}{1-4z},  \n\\end{eqnarray*}\nwhich is separable.  Separating the variables and integrating,\n\n<\/p>\n\n\n\n<p>\n\t\t\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\int \\frac{4z^{2}-1}{1-4z}dz &amp;= &amp; \\int \\frac{1}{x}dx  \\\\\n\t\t\t\t\\int \\left( \\frac{-3\/2}{2z+1} + \\frac{-1\/2}{2z-1} \\right) dz &amp; = &amp;\n\t\t\t\t\\int \\frac{1}{x}dx \\\\\n\t\t\t\t-\\frac{3}{4} \\ln |2z+1| &#8211; \\frac{1}{4} \\ln |2z-1| &amp; = &amp; \\ln |x| +\n\t\t\t\tC_{1}\\\\\n\t\t\t\t3 \\ln |2z+1| + \\ln |2z-1| &amp; = &amp; -4\\ln |x| + C_{2}\\\\\n\t\t\t\t\\ln \\left| (2z+1)^{3}(2z-1)x^{4} \\right| &amp; = &amp; C_{2}\\\\\n\t\t\t\te^{\\ln \\left| (2z+1)^{3}(2z-1)x^{4} \\right|} &amp; = &amp; e^{C_{2}}\\\\\n\t\t\t\t\\left| (2z+1)^{3}(2z-1)x^{4} \\right| &amp; = &amp; C\\\\\n\t\t\t\t\\left( 2\\frac{y}{x}+1  \\right) ^{3} \\left( 2\\frac{y}{x}-1 \\right)x^{4}\n\t\t\t\t&amp; = &amp; C\\\\\n\t\t\t\t(2y+x)^{3}(2y-x) &amp; = &amp; C.\n\t\t\t\\end{eqnarray*}\n<\/p>\n\n\n\n<p>\n\t\t\t\tUsing partial fractions.\n<\/p>\n\n\n\n<p>\n\t\t\t\tMultiplying through by $-4$ and relabelling $-4C_{1}$ as $C_{2}$.\n<\/p>\n\n\n\n<p>\n\t\t\t\tReturning to the original variables using $z=y\/x$.\n<\/p>\n\n\n\n<p>\nThe general solution, $(2y+x)^{3}(2y-x)  =  C$, is written implicitly.\n\n<\/p>\n\n\n<style>.kt-accordion-id_1cf274-2c .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_1cf274-2c .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_1cf274-2c > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > 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.kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_1cf274-2c:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_1cf274-2c:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_1cf274-2c:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_1cf274-2c:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_1cf274-2c:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_1cf274-2c .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_1cf274-2c .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_1cf274-2c kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_44449d-4a\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">View Solution<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/odetut3.gif?resize=485%2C348&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-374\" width=\"485\" height=\"348\"\/><\/figure><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Finding an Integral for an Exact Equation<\/h4>\n\n\n\n<p>\n\nAn ODE $N(x,y)y&#8217; + M(x,y) = 0$ is an <b>exact<\/b> equation if\n$\\frac{\\partial N}{\\partial x} = \\frac{\\partial M}{\\partial y}$ in a\nregion of the $xy$-plane.  If we can find a function $H(x,y)$ for\nwhich $\\frac{\\partial H}{\\partial x}= M$ and $\\frac{\\partial\nH}{\\partial y}= N$, then $H(x,y)$ is called an <b>integral<\/b> of the\nODE and $H(x,y) = C$ is the general solution of the original ODE.\n\n<\/p>\n\n\n\n<p>To find $H(x,y)$, note that $$ H(x,y) = \\int M(x,y)dx + g(y) $$ for some $g(y)$ since $\\frac{\\partial H}{\\partial x}= M(x,y)$.  To find $g(y)$, calculate $$ \\frac{\\partial H}{\\partial y}=\\frac{\\partial}{\\partial y} \\left[ \\int M(x,y)dx \\right] + g'(y) $$ and set it equal to $N(x,y)$.  Solve for $g'(x,y)$ ,which will be independent of $ x $, and integrate with respect to $y$ to obtain $g(y)$, and so $H(x,y)$, explicitly.  Notice that our solution $H(x,y) = C$ is written in implicit form.  Alternatively, we can start with $H(x,y) = \\int N(x,y)dy + h(x)$ for some $h(x)$ and proceed accordingly.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Example<\/h4>\n\n\n\n<p>\n\nThe ODE $\\left( 2yx^{2} + 4 \\right) \\frac{dy}{dx} + \\left( 2y^{2}x -3\n\\right) = 0$ is exact, since for $N(x,y) = 2yx^{2} + 4$ and $M(x,y) =\n2y^{2}x -3$,\n$$\n\\frac{\\partial N}{\\partial x}=4xy=\\frac{\\partial M}{\\partial y}.\n$$\nThus, there exists an integral $H(x,y)$ for which\n$$\n\\frac{\\partial H}{\\partial x}= 2y^{2}x-3 \\mathrm{~and~}\n\\frac{\\partial H}{\\partial y}= 2yx^{2}+4.\n$$\nFrom the first of these,\n\\begin{eqnarray*}\n\tH(x,y) &amp;= &amp; \\int \\left( 2y^{2}x -3 \\right) dx + g(y) \\\\\n\tH(x,y) &amp;= &amp; y^{2}x^{2}-3x+g(y) .\n\\end{eqnarray*}\nThen $\\frac{\\partial H}{\\partial y}=2yx^{2} + g'(y)=2yx^{2}+4$, so\n$g'(y)=4$.\n\n<\/p>\n\n\n\n<p>\nIntegrating, $g(y)= 4y$, so $H(x,y)= y^{2}x^{2}-3x+4y$.\n\n<\/p>\n\n\n\n<p> The general solution is given implicitly by  $$ y^{2}x^{2}-3x+4y =C. $$ <\/p>\n\n\n<style>.kt-accordion-id_3a03bb-54 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_3a03bb-54 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_3a03bb-54 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap 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.kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_3a03bb-54 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_3a03bb-54 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_3a03bb-54:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_3a03bb-54 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_3a03bb-54 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_3a03bb-54 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_b3f727-be\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">View Solutions<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<div class=\"wp-block-image\"><figure class=\"aligncenter is-resized\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/odetut4.gif?resize=507%2C366&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-375\" width=\"507\" height=\"366\"\/><\/figure><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<p>\n<\/p><h3>Key Concepts<\/h3>\n<p>\n<\/p><\/center>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ2410\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p>\n\n\n","protected":false},"excerpt":{"rendered":"<p>Consider the first-order ODE, which is when the highest derivative appearing in the equation is a first derivative. $y&#8217;=f(t,y)$, $y(t_{0})=y_{0}$ Does $y&#8217;=f(t,y)$, $y(t_{0})=y_{0}$ have a solution? If so, can we find a formula for the solution? The first question is easily addressed: Existence and Uniqueness Theorem Suppose that $f(t,y)$ and $\\frac{\\partial f(t,y)}{\\partial y}$ are continumous&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":63,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-233","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/233","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=233"}],"version-history":[{"count":17,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/233\/revisions"}],"predecessor-version":[{"id":1238,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/233\/revisions\/1238"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/63"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=233"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=233"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}