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</ol> <article id="main-article"> <p><img alt="" src="../../files/integration/differential-equations/integrating-factor/main-1.png" style="float: left; width: 100px; height: 100px;">On this page we will look at another type of differential equations: linear differential equations in the form y' + P(x)y=Q(x) which can be solved by using an integrating factor. There are many applications of this type of differential equation. For example, in electrical engineering, we can solve Kirchhoff's Laws to model electrical circuits. To be successful with this topic, you will need to have strong foundations in the following areas: integration techniques (especially in the form <span class="math-tex">\(\int\frac{f'(x)}{f(x)} \mathrm{d}x\)</span>) and manipulation of logarithms and exponents. You can get some practice in these areas if you complete all the quizzes on this page before attempting the exam-style questions.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn about </p> <ul> <li>solving <span class="math-tex">\(\large \frac{\text{d}y}{\text{d}x}+P(x)y=Q(x)\)</span> </li> <li>using the integrating factor <span class="math-tex">\(I=e^{\int P(x)\mathrm{d}x}\)</span></li> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div> <p>In order to use the integrating factor method, the differential equation needs to be in the form <span class="math-tex">\(\large \frac{\text{d}y}{\text{d}x}+P(x)y=Q(x)\)</span>. In the following videos we look at how the method works.</p> <div class="panel panel-yellow panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Easier Example </p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="904"> <p>In the following video we will look at an easier question that requires the integrating factor method to solve it. In particular, we will look at how the method works and why it works. You will see that you need to be confident in quite a few areas of the HL course to be able to carry it out. In this case, we need to use integration by recognition and integration by parts.</p> <hr class="hidden-separator"> <p>Find the general solution to the differential equation</p> <p style="text-align: center;"><span class="math-tex">\(\large \frac{\text{d}y}{\text{d}x}+\frac{y}{x}=\sin x\)</span></p> <p>in the form <span class="math-tex">\(\large y=f(x)\)</span></p> <hr class="hidden-separator"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/589232581"></iframe></div> <h4><span></span><span></span>Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Print from <a href="../../files/integration/differential-equations/integrating-factor/de_intfactor1_videonotes.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/differential-equations/integrating-factor/de_intfactor1_videonotes.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Difficult Example</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="905"> <p>In the following video we look at how we can find the particular solution to a differential equation using an integrating factor. This example is more challenging, since we have to re-arrange the differential equation to put it in the correct form and the integration is more difficult.</p> <hr class="hidden-separator"> <p>Find the particular solution to the differential equation</p> <p style="text-align: center;"><span class="math-tex">\(\large \cos x\frac{\text{d}y}{\text{d}x}+{y}\ {\sin x}=\sin 2x\)</span></p> <p>given that <span class="math-tex">\(\large y(0)=2\)</span></p> <hr class="hidden-separator"> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/593154555"></iframe></div> <h4><span></span><span></span>Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Print from <a href="../../files/integration/differential-equations/integrating-factor/de_intfactor2_videonotes.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/integration/differential-equations/integrating-factor/de_intfactor2_videonotes.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/integration/differential-equations/integrating-factor/revision-notes_de_integrating_factor.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/integration/differential-equations/integrating-factor/revision-notes_de_integrating_factor.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <div> <p>To solve differential equations with an integrating factor, we are often required to manipulate <strong>exponents of logarithmic functions</strong>. This will give you practice in that skill</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#b22412df"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="b22412df"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Manipulating exponents of logarithmic functions <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-589-2109" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Simplify <span class="math-tex">\(e^{\ln x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> x</label></p><p><label class="radio"><input type="radio"> -x</label></p><p><label class="radio"><input type="radio"> x²</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\({1} \over x\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(e^x\)</span> and <span class="math-tex">\(\ln x\)</span> are inverse functions, so the exponential function will undo the logarithmic function</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Simplify <span class="math-tex">\(e^{\ln x^2}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\({1} \over x^2\)</span></label></p><p><label class="radio"><input type="radio"> 2x</label></p><p><label class="radio"><input class="c" type="radio"> x²</label></p><p><label class="radio"><input type="radio"> x</label></p></div><div class="q-explanation"><p><span class="math-tex">\(e^x\)</span> and <span class="math-tex">\(\ln x\)</span> are inverse functions, so the exponential function will undo the logarithmic function</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Simplify <span class="math-tex">\(e^{3\ln x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> x</label></p><p><label class="radio"><input type="radio"> 3x</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\({1} \over x\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(x^3\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(e^{3\ln x}=e^{\ln x^3}\)</span></p><p><span class="math-tex">\(e^x\)</span><span tabindex="-1"></span> and <span class="math-tex">\(\ln x\)</span> are inverse functions, so the exponential function will undo the logarithmic function</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Simplify <span class="math-tex">\(e^{-\ln x}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> x</label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{1}{x}\)</span></label></p><p><label class="radio"><input type="radio"> -x</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-\frac{1}{x}\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(e^{-\ln x}=e^{\ln x^{-1}}=e^{\ln \frac{1}{x}}\)</span></p><p><span class="math-tex">\(e^x\)</span><span tabindex="-1"></span> and <span class="math-tex">\(\ln x\)</span> are inverse functions, so the exponential function will undo the logarithmic function</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Find the correct simplifications of the following questions</p><p><br> <span class="q-text-draggable draggable" draggable="true">cosec(x)</span> <span class="q-text-draggable draggable" draggable="true">2sin(x)</span> <span class="q-text-draggable draggable" draggable="true">cosec²(x)</span> <span class="q-text-draggable draggable" draggable="true">sin²(x)</span> <span class="q-text-draggable draggable" draggable="true">sin(x)</span> <span class="q-text-draggable draggable" draggable="true">-sin(x)</span> </p></div><div class="q-answer"><p>a) <span class="math-tex">\(e^{\ln(\sin x)}\)</span> = <input type="text" style="height: auto;" data-c="sin(x)"> <span class="review"></span> </p><p>b) <span class="math-tex">\(e^{-\ln(\sin x)}\)</span> = <input type="text" style="height: auto;" data-c="cosec(x)"> <span class="review"></span> </p><p>c) <span class="math-tex">\(e^{2\ln(\sin x)}\)</span> = <input type="text" style="height: auto;" data-c="sin²(x)"> <span class="review"></span> </p><p>d) <span class="math-tex">\(e^{-2\ln(\sin x)}\)</span> = <input type="text" style="height: auto;" data-c="cosec²(x)"> <span class="review"></span> </p></div><div class="q-explanation"><p><span class="math-tex">\(e^x\)</span><span tabindex="-1"></span> and <span class="math-tex">\(\ln x\)</span> are inverse functions, so the exponential function will undo the logarithmic function</p><p>b) <span class="math-tex">\(e^{-\ln(\sin x)}=e^{\ln(\sin x)^{-1}}=e^{\ln(\frac{1}{\sin x})}=cosecx\)</span></p><p>c) <span class="math-tex">\(e^{2\ln(\sin x)}=e^{\ln(\sin x)^2}=sin²x\)</span></p><p>d) <span class="math-tex">\(e^{-2\ln(\sin x)}=e^{\ln(\sin x)^{-2}}=e^{\ln\frac{1}{(\sin x)^2}}=cosec²x\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Simplify <span class="math-tex">\(e^{\ln (1+x)}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> ln(1+x)</label></p><p><label class="radio"><input type="radio"> x</label></p><p><label class="radio"><input class="c" type="radio"> 1 + x</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(e^{1+x}\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(e^x\)</span><span tabindex="-1"></span> and <span class="math-tex">\(\ln x\)</span> are inverse functions, so the exponential function will undo the logarithmic function</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Simplify <span class="math-tex">\(e^{-3ln(1+x)}\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(-(1+x)^3\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{3}{1+x}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{1}{(1+x)^3}\)</span></label></p><p><label class="radio"><input type="radio"> -3(1+x)</label></p></div><div class="q-explanation"><p><span class="math-tex">\(e^{-3ln(1+x)}=e^{ln(1+x)^{-3}}\)</span></p><p><span class="math-tex">\(e^x\)</span><span tabindex="-1"></span> and <span class="math-tex">\(\ln x\)</span> are inverse functions, so the exponential function will undo the logarithmic function</p><p><span class="math-tex">\(=(1+x)^{-3}=\frac{1}{(1+x)^3}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Simplify <span class="math-tex">\(e^{-\frac{1}{2}ln(1-x^2)}\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{1}{\sqrt{1-x^2}}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{1-x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{2(1-x^2)}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-\frac{1}{2}(1-x^2)\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(e^{-\frac{1}{2}ln(1-x^2)}=e^{ln(1-x^2)^{-\frac{1}{2}}}\)</span></p><p><span class="math-tex">\(e^x\)</span><span tabindex="-1"></span> and <span class="math-tex">\(\ln x\)</span> are inverse functions, so the exponential function will undo the logarithmic function</p><p><span class="math-tex">\(=(1-x^2)^{-\frac{1}{2}}=\frac{1}{\sqrt{1-x^2}}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Often differential equations with an integrating factor require us to carry out integrations in the form <span class="math-tex">\(\large\int\frac{f'(x)}{f(x)} \mathrm{d}x=\ln |f(x)|+C\)</span></p> <p>The following quiz wil help you practise these type of integrals</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#5a9fe73e"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="5a9fe73e"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Integrals in form f'(x)/f(x) <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-590-2109" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\int\frac{1}{2x} \mathrm{d}x\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\ln |x|+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{1}{2}\ln |x|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\ln |2x|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{2x}+C\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\int\frac{1}{2x} \mathrm{d}x=\frac{1}{2} \int\frac{1}{x} \mathrm{d}x=\frac{1}{2}\ln |x|+C\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\int\frac{2x}{x^2+1} \mathrm{d}x\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{2}\ln|x^2+1|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2\ln |x^2+1|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{x^2+1}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\ln |x^2+1|+C\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac{\mathrm{d}}{\mathrm{d}x}(x^2+1)=2x\)</span></p><p>Hence integral is in the form <span class="math-tex">\(\int\frac{f'(x)}{f(x)} \mathrm{d}x=\ln |f(x)|+C\)</span></p><p><span class="math-tex">\(\int\frac{2x}{x^2+1} \mathrm{d}x=\ln |x^2+1|+C\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\int\frac{\cos x}{\sin x} \mathrm{d}x\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\tan x+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-\ln|\sin x|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\ln|\cos x|+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\ln |\sin x|+C\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac{\mathrm{d}}{\mathrm{d}x}(\sin x)=\cos x\)</span></p><p>Hence integral is in the form <span class="math-tex">\(\int\frac{f'(x)}{f(x)} \mathrm{d}x=\ln |f(x)|+C\)</span></p><p><span class="math-tex">\(\int\frac{\cos x}{\sin x} \mathrm{d}x=\ln |\sin x|+C\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\int\tan x \mathrm{d}x\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\ln|\cos x|+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(-\ln |\cos x|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\sec² x+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\ln|\sin x|+C\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\int\tan x \mathrm{d}x=\int\frac{\sin x}{\cos x} \mathrm{d}x=-\int\frac{-\sin x}{\cos x} \mathrm{d}x\)</span></p><p><span class="math-tex">\(\frac{\mathrm{d}}{\mathrm{d}x}(\cos x)=-\sin x\)</span></p><p>Hence integral is in the form <span class="math-tex">\(\int\frac{f'(x)}{f(x)} \mathrm{d}x=\ln |f(x)|+C\)</span></p><p><span class="math-tex">\(-\int\frac{-\sin x}{\cos x} \mathrm{d}x=-\ln |\cos x|+C\)</span></p><p>We could also write this answer as <span class="math-tex">\(\ln |\sec x|+C\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\int\frac{4}{x} \mathrm{d}x\)</span></p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(-\frac{4}{x^2}+C\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(4\ln |x|+C\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\ln |x|+C\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\ln |x^4|+C\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\int\frac{4}{x} \mathrm{d}x=4 \int\frac{1}{x} \mathrm{d}x=4\ln |x|+C\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\int\frac{1}{1-x} \mathrm{d}x\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\ln|1-x|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(-\ln|1-x²|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{1-x^2}+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(-\ln|1-x|+C\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\int\frac{1}{1-x} \mathrm{d}x=-\int\frac{-1}{1-x} \mathrm{d}x\)</span></p><p><span class="math-tex">\(\frac{\mathrm{d}}{\mathrm{d}x}(1-x)=-1\)</span></p><p>Hence integral is in the form <span class="math-tex">\(\int\frac{f'(x)}{f(x)} \mathrm{d}x=\ln |f(x)|+C\)</span></p><p><span class="math-tex">\(-\int\frac{-1}{1-x} \mathrm{d}x=-\ln|1-x|+C\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\int\frac{x}{x^2+1} \mathrm{d}x\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(x^2+1+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{1}{2}\ln|x^2+1|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2\ln|x^2+1|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\ln|x^2+1|+C\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\int\frac{x}{x^2+1} \mathrm{d}x=\frac{1}{2}\int\frac{2x}{x^2+1} \mathrm{d}x\)</span></p><p><span class="math-tex">\(\frac{\mathrm{d}}{\mathrm{d}x}(x^2+1)=2x\)</span></p><p>Hence integral is in the form <span class="math-tex">\(\int\frac{f'(x)}{f(x)} \mathrm{d}x=\ln |f(x)|+C\)</span></p><p><span class="math-tex">\(\frac{1}{2}\int\frac{2x}{x^2+1} \mathrm{d}x=\frac{1}{2}\ln|x^2+1|+C\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\int\frac{x+3}{x^2+6x+1} \mathrm{d}x\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(2\ln|x^2+6x+1|+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{1}{2}\ln|x^2+6x+1|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\ln |x+3|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\ln|x^2+6x+1|+C\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\int\frac{x+3}{x^2+6x+1} \mathrm{d}x=\frac{1}{2}\int\frac{2x+6}{x^2+6x+1} \mathrm{d}x\)</span></p><p><span class="math-tex">\(\frac{\mathrm{d}}{\mathrm{d}x}(x^2+6x+1)=2x+6\)</span></p><p>Hence integral is in the form <span class="math-tex">\(\int\frac{f'(x)}{f(x)} \mathrm{d}x=\ln |f(x)|+C\)</span></p><p><span class="math-tex">\(\frac{1}{2}\int\frac{2x+6}{x^2+6x+1} \mathrm{d}x=\frac{1}{2}\ln|x^2+6x+1|+C\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\int\frac{x^3}{1+x^4} \mathrm{d}x\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\ln|1+x^4|+C\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{1}{4}\ln|1+x^4|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(4\ln|1+x^4|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(4\ln|x^3|+C\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\int\frac{x^3}{1+x^4} \mathrm{d}x=\frac{1}{4}\int\frac{4x^3}{1+x^4} \mathrm{d}x\)</span></p><p><span class="math-tex">\(\frac{\mathrm{d}}{\mathrm{d}x}(1+x^4)=4x^3\)</span></p><p>Hence integral is in the form <span class="math-tex">\(\int\frac{f'(x)}{f(x)} \mathrm{d}x=\ln |f(x)|+C\)</span></p><p><span class="math-tex">\(\frac{1}{4}\int\frac{4x^3}{1+x^4} \mathrm{d}x=\frac{1}{4}\ln|1+x^4|+C\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <span class="math-tex">\(\large\int\frac{1}{x\ln x} \mathrm{d}x\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\ln|\ln x|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{x}+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\ln|x\ln x|+C\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{x^2(\ln x)^2}+C\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\int\frac{1}{x\ln x} \mathrm{d}x=\int\frac{\frac{1}{x}}{\ln x} \mathrm{d}x\)</span></p><p><span class="math-tex">\(\frac{\mathrm{d}}{\mathrm{d}x}(\ln x)=\frac{1}{x}\)</span></p><p>Hence integral is in the form <span class="math-tex">\(\int\frac{f'(x)}{f(x)} \mathrm{d}x=\ln |f(x)|+C\)</span></p><p><span class="math-tex">\(\int\frac{\frac{1}{x}}{\ln x} \mathrm{d}x=\ln|\ln x|+C\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>The integrating factor, <em><strong><span class="math-tex">\(I\)</span> </strong></em>for the differential equation in the form <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\)</span> can be found by evaluating <span class="math-tex">\(\large I=e^{\int P(x) \mathrm{d}x}\)</span></p> <p>The following quiz gives you some practice in finding the integrating factor</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#77f1c762"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="77f1c762"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Integrating factor for differential equations <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-591-2109" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The integrating factor, <em><strong><span class="math-tex">\(I\)</span> </strong></em>for the differential equation in the form <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\)</span> can be found by evaluating <span class="math-tex">\(\large I=e^{\int P(x) \mathrm{d}x}\)</span></p><hr class="hidden-separator"><p>What is the integrating factor for the differential equation <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}+2y=e^x\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(e^x\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2x\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(e^{x^2}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(e^{2x}\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(I=e^{\int 2 \ \mathrm{d}x}\\ I=e^{2x}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The integrating factor, <em><strong><span class="math-tex">\(I\)</span> </strong></em>for the differential equation in the form <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\)</span> can be found by evaluating <span class="math-tex">\(\large I=e^{\int P(x) \mathrm{d}x}\)</span></p><hr class="hidden-separator"><p>What is the integrating factor for the differential equation <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}+\frac{y}{x}=\sin x\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(x\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\ln x\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(e^{-\frac{1}{x^2}}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(e^x\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(I=e^{\int \frac{1}{x} \ \mathrm{d}x}\\ I=e^{\ln x}\\ I=x\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The integrating factor, <em><strong><span class="math-tex">\(I\)</span> </strong></em>for the differential equation in the form <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\)</span> can be found by evaluating <span class="math-tex">\(\large I=e^{\int P(x) \mathrm{d}x}\)</span></p><hr class="hidden-separator"><p>What is the integrating factor for the differential equation <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}+\frac{2}{x}y=x\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{2}x\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(2x\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(e^{2x}\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(x^2\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(I=e^{\int \frac{2}{x} \mathrm{d}x}\\ I=e^{2\ln x}\\ I=e^{\ln x^2}\\ I=x^2\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The integrating factor, <em><strong><span class="math-tex">\(I\)</span> </strong></em>for the differential equation in the form <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\)</span> can be found by evaluating <span class="math-tex">\(\large I=e^{\int P(x) \mathrm{d}x}\)</span></p><hr class="hidden-separator"><p>What is the integrating factor for the differential equation <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}-\frac{x^2y}{1+x^3}=x^2\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(-(1+x^3)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{{1+x^3}}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(1+x^3\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{1}{\sqrt[ 3]{1+x^3}}\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(I=e^{\int -\frac{x^2}{1+x^3} \mathrm{d}x}\\ I=e^{-\int \frac{x^2}{1+x^3} \mathrm{d}x}\\ I=e^{-\frac{1}{3}\int \frac{3x^2}{1+x^3} \mathrm{d}x}\\ I=e^{-\frac{1}{3}\ln |1+x^3|}\\ I=e^{\ln (1+x^3)^{-\frac{1}{3}}}\\ I= (1+x^3)^{-\frac{1}{3}}\\ I=\frac{1}{\sqrt[ 3]{1+x^3}}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The integrating factor, <em><strong><span class="math-tex">\(I\)</span> </strong></em>for the differential equation in the form <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\)</span> can be found by evaluating <span class="math-tex">\(\large I=e^{\int P(x) \mathrm{d}x}\)</span></p><hr class="hidden-separator"><p>What is the integrating factor for the differential equation <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}+y \tan x=\sin x\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(\arccos x\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{1}{\cos x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(e^{\sec^2x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\cos x\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(I=e^{\int \tan x \ \mathrm{d}x}\\ I=e^{\int \frac{\sin x}{\cos x} \mathrm{d}x}\\ I=e^{-\int \frac{-\sin x}{\cos x} \mathrm{d}x}\\ I=e^{-\ln \cos x}\\ I=e^{\ln (\cos x)^{-1}}\\ I=(\cos x)^{-1}\\ I=\frac{1}{\cos x}\\ I=\sec x\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The integrating factor, <em><strong><span class="math-tex">\(I\)</span> </strong></em>for the differential equation in the form <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\)</span> can be found by evaluating <span class="math-tex">\(\large I=e^{\int P(x) \mathrm{d}x}\)</span></p><hr class="hidden-separator"><p>What is the integrating factor for the differential equation <span class="math-tex">\(\large x\frac{\mathrm{d}y}{\mathrm{d}x}-y=x^3\)</span></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(-x^2\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{1}{x^2}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(x^2\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{x}\)</span></label></p></div><div class="q-explanation"><p>We need to put the differential equation in the correct form</p><p><span class="math-tex">\( x\frac{\mathrm{d}y}{\mathrm{d}x}-y=x^3\\ \frac{\mathrm{d}y}{\mathrm{d}x}-\frac{y}{x}=x^2\)</span></p><p><span class="math-tex">\(I=e^{\int -\frac{2}{x} \mathrm{d}x}\\ I=e^{-2\ln x}\\ I=e^{\ln x^{-2}}\\ I=x^{-2}\\ I=\frac{1}{x^2}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The integrating factor, <em><strong><span class="math-tex">\(I\)</span> </strong></em>for the differential equation in the form <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\)</span> can be found by evaluating <span class="math-tex">\(\large I=e^{\int P(x) \mathrm{d}x}\)</span></p><hr class="hidden-separator"><p>What is the integrating factor for the differential equation <span class="math-tex">\(\large (1+x)\frac{\mathrm{d}y}{\mathrm{d}x}+y=1+x\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(1+x\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{-1}{(1+x)^2}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{1+x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\ln (1+x)\)</span></label></p></div><div class="q-explanation"><p>We need to put the differential equation in the correct form</p><p><span class="math-tex">\((1+x)\frac{\mathrm{d}y}{\mathrm{d}x}+y=1+x\\ \frac{\mathrm{d}y}{\mathrm{d}x}+\frac{1}{1+x}y=1\)</span></p><p><span class="math-tex">\(I=e^{\int \frac{1}{1+x} \mathrm{d}x}\\ I=e^{\ln (1+x)}\\ I=1+x\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The integrating factor, <em><strong><span class="math-tex">\(I\)</span> </strong></em>for the differential equation in the form <span class="math-tex">\(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\)</span> can be found by evaluating <span class="math-tex">\(\large I=e^{\int P(x) \mathrm{d}x}\)</span></p><hr class="hidden-separator"><p>What is the integrating factor for the differential equation <span class="math-tex">\(\large\sin x\frac{\mathrm{d}y}{\mathrm{d}x}+y \cos x=\mathrm{cosec} x\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\sin x\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(e^{\cot x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(e^{\sin x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\mathrm{cosec}x\)</span></label></p></div><div class="q-explanation"><p>We need to put the differential equation in the correct form</p><p><span class="math-tex">\(\sin x\frac{\mathrm{d}y}{\mathrm{d}x}+y \cos x=\mathrm{cosec} x\\ \frac{\mathrm{d}y}{\mathrm{d}x}+ \frac{\cos x}{\sin x}y=\frac{1}{\sin^2x}\)</span></p><p><span class="math-tex">\( I=e^{\int \frac{\cos x}{\sin x} \mathrm{d}x}\\ I=e^{\ln \sin x}\\ I=\sin x\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="897"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Consider the following first order differential equation</p> <p><span class="math-tex">\(\large x\frac{\text{d}y}{\text{d}x}+3y=\frac{lnx}{x}\)</span></p> <p>a) Show that <em><strong>x<sup>3</sup></strong></em> is the integrating factor for this differential equation</p> <p>b) Hence, find the general solution of this differential equation in the form <em><strong>y = f(x)</strong></em></p> <h4>Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>a) The integrating factor, <span class="math-tex">\(I=e^{\int \frac{3}{x}dx}\)</span></p> <p>b) We should recognise the product rule for differentiation <img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor1hint.png" style="width: 250px; height: 43px;"></p> <p>We should use integration by parts to integrate xlnx</p> </section> <h4>Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor1.png" style="width: 600px; height: 259px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor1b.png" style="width: 928px; height: 647px;"></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="898"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Consider the following differential equation <span class="math-tex">\(\large \frac{\text{d}y}{\text{d}x}+ytanx=secx\)</span></p> <p>a) Using a suitable integrating factor show that the differential equation can be written as <span class="math-tex">\(\large \frac{y}{cosx}=\int sec^²x {dx}\)</span></p> <p>b) Given that (0 , 2) lies on the curve, show that the particular solution of the differential equation is<strong><em> <span class="math-tex">\(\large y=sinx+2cosx\)</span></em></strong></p> <h4>Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) This is a differential equation in the form <span class="math-tex">\(\large \frac{\text{d}y}{\text{d}x}+P(x)y=Q(x)\)</span></p> <p>We use the integrating factor <span class="math-tex">\(I=e^{\int tanxdx }\)</span></p> <p><content> </content>You will need to use integration by substitution or recognition to integrate <em><strong>tanx</strong></em></p> <p>We should recognise the product rule for differentiation <img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor2hint.png" style="width: 300px; height: 61px;"></p> </section> <h4>Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor2a.png" style="width: 600px; height: 256px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor2b.png" style="width: 600px; height: 553px;"></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="902"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Consider the following differential equation <span class="math-tex">\(\large \sin x\frac{\text{d}y}{\text{d}x}+y\cos x=2\sin^2 x\)</span></p> <p>a) Show that the integrating factor of this differential equation is <span class="math-tex">\(\large \sin x\)</span></p> <p>b) Solve the differential equation, giving your answer in the form <span class="math-tex">\(y=f(x)\)</span></p> <h4>Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>a) This is a differential equation in the form <span class="math-tex">\(\large \frac{\text{d}y}{\text{d}x}+P(x)y=Q(x)\)</span></p> <p>We use the integrating factor <span class="math-tex">\(\large I=e^{\int \frac{\cos x}{\sin x}dx }\)</span></p> <p><content> </content>You will need to use integration by substitution or recognition for this integration</p> <p>b) We should recognise the product rule for differentiation <img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactorhint.png" style="width: 400px; height: 70px;"></p> </section> <h4>Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor3a.png" style="width: 600px; height: 348px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor3b.png" style="width: 600px; height: 310px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_integratingfactor3c.png" style="width: 600px; height: 118px;"></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div 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