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</ol> <article id="main-article"> <p><img alt="" src="../../files/integration/differential-equations/separablemain.jpg" style="float: left; width: 100px; height: 100px;">If you want to describe the world around you, be it the forces acting on a body, the growth of a virus or the temperature of the coffee in your cup, you will be dealing with differential equations. On this page we will look at the simplest type: differential equations in which we can separate the variables. To be successful with the topic on this page, you will need to have strong foundations in the following areas: integration techniques, manipulation of logarithms and exponents and partial fractions. It is recommended that you refresh your techniques in those areas if you are not feeling 100% confident.</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 to</p> <ul> <li>recognise differential equations with separable variables - in the form <span class="math-tex">\(\frac{dy}{dx}=f(x)g(y)\)</span></li> <li>solve these differential equations using integration</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>The following videos will help you understand all the concepts from this page</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>Example 1 - Finding a General Solution</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="901"> <p>In the following video we look at how we can solve separable differential equations</p> <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/557750454"></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/de_separable1_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/de_separable1_videonotes.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <p> </p> <div class="panel panel-orange panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Example 2 - Finding a Specific Solution</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="900"> <p>In the following video, we will see an example of finding a particular solution to a differential equation. The variables in this differential equation are separable</p> <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/557545020"></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/de_separable2_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/de_separable2_videonotes.pdf" width="640"></iframe></p> </section> </div> <p> </p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <p> </p> </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 style="text-align: center;"><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/integration/differential-equations/revision-notes_de_variables_separable.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/integration/differential-equations/revision-notes_de_variables_separable.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, we are often required to manipulate exponents and logarithms. Here is a quiz that will give you some practice in what you need</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#46be168f"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="46be168f"> <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 and logs <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="11-582-2107" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is equivalent to <span class="math-tex">\(e^{x+y}\)</span>?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(ye^x\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(e^x+e^y\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(e^x \cdot e^y\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(e^{xy}\)</span></span></label> </p></div><div class="q-explanation"><p>From the first law of indices <span class="math-tex">\(a^{x+y}=a^x \cdot a^y\)</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>Which if the following is equivalent to <span class="math-tex">\(e^{x-y}\)</span></p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\large e^x-e^y\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\large e^{\frac{x}{y}}\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\large\frac{e^x} {e^y}\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\large e^x \cdot e^{-y}\)</span></label></p></div><div class="q-explanation"><p>From the first law of indices <span class="math-tex">\(a^{x-y}=a^{x+(-y)}=a^x \cdot a^{-y}\)</span></p><p>From the second law of indices <span class="math-tex">\(a^{x-y}=a^x \div a^y\)</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>Select the correct answers to the following</p></div><div class="q-answer"><p> <span class="q-text-draggable draggable" draggable="true">0</span> <span class="q-text-draggable draggable" draggable="true">1</span> <span class="q-text-draggable draggable" draggable="true">x</span> <span class="q-text-draggable draggable" draggable="true">2x</span> <span class="q-text-draggable draggable" draggable="true">x²</span> </p><p><span class="math-tex">\(\large e^x \cdot e^{-x}=\)</span> <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p><p><span class="math-tex">\(\large e^{lnx}=\)</span> <input type="text" style="height: auto;" data-c="x"> <span class="review"></span></p><p><span class="math-tex">\(\large ln(e^x)=\)</span> <input type="text" style="height: auto;" data-c="x"> <span class="review"></span></p></div><div class="q-explanation"></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>Select the correct answers to the following</p></div><div class="q-answer"><p> <span class="q-text-draggable draggable" draggable="true">0</span> <span class="q-text-draggable draggable" draggable="true">1</span> <span class="q-text-draggable draggable" draggable="true">x</span> <span class="q-text-draggable draggable" draggable="true">2x</span> <span class="q-text-draggable draggable" draggable="true">x²</span> </p><p><span class="math-tex">\(\large e^{lnx²}=\)</span> <input type="text" style="height: auto;" data-c="x²"> <span class="review"></span></p><p><span class="math-tex">\(\large e^{2lnx}=\)</span> <input type="text" style="height: auto;" data-c="x²"> <span class="review"></span></p><p><span class="math-tex">\(\large ln(e^{x²})=\)</span> <input type="text" style="height: auto;" data-c="x²"> <span class="review"></span></p></div><div class="q-explanation"></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>Select the correct answers to the following</p></div><div class="q-answer"><p> <span class="q-text-draggable draggable" draggable="true">cosx</span> <span class="q-text-draggable draggable" draggable="true">cos²x</span> <span class="q-text-draggable draggable" draggable="true">2cosx</span> <span class="q-text-draggable draggable" draggable="true">ln(cosx)</span></p><p><span class="math-tex">\(\large e^{ln(cosx)}=\)</span> <input type="text" style="height: auto;" data-c="cosx"> <span class="review"></span></p><p><span class="math-tex">\(\large e^{2ln(cosx)}=\)</span> <input type="text" style="height: auto;" data-c="cos²x"> <span class="review"></span></p><p><span class="math-tex">\(\large ln(e^{2cosx})=\)</span> <input type="text" style="height: auto;" data-c="cos²x"> <span class="review"></span></p></div><div class="q-explanation"></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>When giving the general solution to differential equation, we have to manipulate arbitrary constants. Try this quiz to ensure that you know how to do this</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#eb7bca4a"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="eb7bca4a"> <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%;"> Dealing with Arbitrary Constants of Integration <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="11-583-2107" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Following an integration, the result is simplified. Decide whether the second statement is a correct simplification of the first one, given that <span class="math-tex">\(\large c\)</span> and <span class="math-tex">\(\large c_1\)</span> are arbitrary constants</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td style="text-align: center;"><span class="math-tex">\(\large y=e^{x+c}\)</span></td><td style="text-align: center;"><span class="math-tex">\(\large y=c_1e^{x}\)</span></td></tr></tbody></table></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Correct!</label></p><p><label class="radio"><input type="radio"> Wrong!</label></p></div><div class="q-explanation"><p>We can write <span class="math-tex">\(\large y=e^{x+c}\)</span> as <span class="math-tex">\(\large y=e^{x}\cdot e^{c}\)</span></p><p><span class="math-tex">\(\large e^c\)</span> is an arbitrary constant</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>Following an integration, the result is simplified. Decide whether the second statement is a correct simplification of the first one, given that <span class="math-tex">\(\large c\)</span> and <span class="math-tex">\(\large c_1\)</span> are arbitrary constants</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td style="text-align: center;"><span class="math-tex">\(\large y=lnx+c\)</span></td><td style="text-align: center;"><span class="math-tex">\(\large y=ln(c_1x)\)</span></td></tr></tbody></table></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Correct!</label></p><p><label class="radio"><input type="radio"> Wrong!</label></p></div><div class="q-explanation"><p>We can write <span class="math-tex">\(\large y=lnx+c\)</span> as <span class="math-tex">\(\large y=lnx+lnc_1\)</span>, since <span class="math-tex">\(lnc_1\)</span> is an arbitrary constant</p><p>This can be simplified to <span class="math-tex">\(\large y=ln(c_1x)\)</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>Following an integration, the result is simplified. Decide whether the second statement is a correct simplification of the first one, given that <span class="math-tex">\(\large c\)</span> and <span class="math-tex">\(\large c_1\)</span> are arbitrary constants</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td style="text-align: center;"><span class="math-tex">\(\large y^2=x+c\)</span></td><td style="text-align: center;"><span class="math-tex">\(\large y=\pm \sqrt{x}+c_1\)</span></td></tr></tbody></table></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Wrong!</label></p><p><label class="radio"><input type="radio"> Correct</label></p></div><div class="q-explanation"><p>The correct simplification is <span class="math-tex">\(\large y=\pm \sqrt{x+c_1}\)</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>Following an integration, the result is simplified. Decide whether the second statement is a correct simplification of the first one, given that <span class="math-tex">\(\large c\)</span> and <span class="math-tex">\(\large c_1\)</span> are arbitrary constants</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td style="text-align: center;"><span class="math-tex">\(\large \sqrt {y}=sinx +c\)</span></td><td style="text-align: center;"><span class="math-tex">\(\large y=sin^2x+c_1\)</span></td></tr></tbody></table></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Wrong!</label></p><p><label class="radio"><input type="radio"> Correct!</label></p></div><div class="q-explanation"><p>The correct simplification is <span class="math-tex">\(\large y=(sinx+c_1)^2\)</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>Following an integration, the result is simplified. Decide whether the second statement is a correct simplification of the first one, given that <span class="math-tex">\(\large c\)</span> and <span class="math-tex">\(\large c_1\)</span> are arbitrary constants</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td style="text-align: center;"><span class="math-tex">\(\large lny=lnx+c\)</span></td><td style="text-align: center;"><span class="math-tex">\(\large y=c_1x\)</span></td></tr></tbody></table></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Correct!</label></p><p><label class="radio"><input type="radio"> Wrong!</label></p></div><div class="q-explanation"><p>We can write <span class="math-tex">\(\large lny=lnx+c\)</span> as <span class="math-tex">\(\large lny=lnx+lnc_1\)</span>, since <span class="math-tex">\(lnc_1\)</span> is an arbitrary constant</p><p>This can be simplified to <span class="math-tex">\(\large lny=ln(c_1x)\)</span></p><p>which can be simplified to <span class="math-tex">\(\large y=c_1x\)</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>Following an integration, the result is simplified. Decide whether the second statement is a correct simplification of the first one, given that <span class="math-tex">\(\large c\)</span> and <span class="math-tex">\(\large c_1\)</span> are arbitrary constants</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td style="text-align: center;"><span class="math-tex">\(\large lny=x^2+c\)</span></td><td style="text-align: center;"><span class="math-tex">\(\large y=c_1e^{x^2}\)</span></td></tr></tbody></table></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Correct!</label></p><p><label class="radio"><input type="radio"> Wrong!</label></p></div><div class="q-explanation"><p>We can write <span class="math-tex">\(\large lny=x^2+c\)</span> as <span class="math-tex">\(\large y=e^{x^2+c}\)</span></p><p>This can be write as <span class="math-tex">\(\large y=e^{x^2}\cdot e^{c}\)</span></p><p><span class="math-tex">\(\large e^c\)</span> is an arbitrary constant</p><p>Therefore, <span class="math-tex">\(\large y=c_1e^{x^2}\)</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>Following an integration, the result is simplified. Decide whether the second statement is a correct simplification of the first one, given that <span class="math-tex">\(\large c\)</span> and <span class="math-tex">\(\large c_1\)</span> are arbitrary constants</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td style="text-align: center;"><span class="math-tex">\(\large y^3=xlnx+c\)</span></td><td style="text-align: center;"><span class="math-tex">\(\large y=\sqrt[3]{xlnx+c_1}\)</span></td></tr></tbody></table></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Correct!</label></p><p><label class="radio"><input type="radio"> Wrong!</label></p></div><div class="q-explanation"><p>This is a correct simplification</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>Following an integration, the result is simplified. Decide whether the second statement is a correct simplification of the first one, given that <span class="math-tex">\(\large c\)</span> and <span class="math-tex">\(\large c_1\)</span> are arbitrary constants</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td style="text-align: center;"><span class="math-tex">\(\large siny=e^{-x}+c\)</span></td><td style="text-align: center;"><span class="math-tex">\(\large y=c_1arcsin(e^{-x})\)</span></td></tr></tbody></table></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Wrong!</label></p><p><label class="radio"><input type="radio"> Correct!</label></p></div><div class="q-explanation"><p>We can simplify to <span class="math-tex">\(\large y=arcsin(e^{-x}+c)\)</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>Following an integration, the result is simplified. Decide whether the second statement is a correct simplification of the first one, given that <span class="math-tex">\(\large c\)</span> and <span class="math-tex">\(\large c_1\)</span> are arbitrary constants</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td style="text-align: center;"><span class="math-tex">\(\large \frac{1}{x}y=lnx+c\)</span></td><td style="text-align: center;"><span class="math-tex">\(\large y=xlnx+c_1\)</span></td></tr></tbody></table></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Wrong!</label></p><p><label class="radio"><input type="radio"> Correct!</label></p></div><div class="q-explanation"><p><span class="math-tex">\(\large \frac{1}{x}y=lnx+c\)</span> can be simplified to <span class="math-tex">\(\large \frac{1}{x}y=lnx+lnc_1\)</span></p><p>which can be written as <span class="math-tex">\(\large \frac{1}{x}y=ln(c_1x)\)</span></p><p>And the general solution is <span class="math-tex">\(\large y=xln(c_1x)\)</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>Following an integration, the result is simplified. Decide whether the second statement is a correct simplification of the first one, given that <span class="math-tex">\(\large c\)</span> and <span class="math-tex">\(\large c_1\)</span> are arbitrary constants</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td style="text-align: center;"><span class="math-tex">\(\large y=ln(x-1)-ln(x+1)+c\)</span></td><td style="text-align: center;"><span class="math-tex">\(\large y=ln\frac{c_1(x-1)}{x+1}\)</span></td></tr></tbody></table></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> Correct!</label></p><p><label class="radio"><input type="radio"> Wrong!</label></p></div><div class="q-explanation"><p>We can write <span class="math-tex">\(\large y=ln(x-1)-ln(x+1)+c\)</span> as <span class="math-tex">\(\large y=ln(x-1)-ln(x+1)+lnc_1\)</span></p><p>And use log laws to simplify this to <span class="math-tex">\(\large y=ln\frac{c_1(x-1)}{x+1}\)</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>Here is a quiz that gets you to practise solving variables separable differential equations</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#8e4a5204"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="8e4a5204"> <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%;"> Differential Equations - Separable <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="11-581-2107" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>How does the following differential equation separate?</p><p style="margin-left: 80px;"><span class="math-tex">\(\large \frac{\text{d}y}{\text{d}x}=e^{x+y}\)</span></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\large \int {e^{-y}}{\text{d}y}=\int e^{x}{\text{d}x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \int {e^{y}}{\text{d}y}=\int e^{x}{\text{d}x}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\large \int {e^{-x}}{\text{d}y}=\int e^{y}{\text{d}x}\)</span></label></p><p><label class="radio"><input type="radio"> The differential equation is not separable</label></p></div><div class="q-explanation"><p>We can re-write the differential equation to be</p><p style="margin-left: 80px;"><span class="math-tex">\(\large \frac{\text{d}y}{\text{d}x}=e^{x}\cdot e^{y}\)</span></p><p style="margin-left: 80px;"><span class="math-tex">\(\large \int \frac{1}{e^{y}}{\text{d}y}=\int e^{x}{\text{d}x}\)</span></p><p style="margin-left: 80px;"><span class="math-tex">\(\large \int {e^{-y}}{\text{d}y}=\int e^{x}{\text{d}x}\)</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>How does the following differential equation separate?</p><p style="margin-left: 80px;"><span class="math-tex">\(\large \frac{\text{d}y}{\text{d}x}=sin({xy})\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\large \int {siny}\ {\text{d}y}=\int sinx\ {\text{d}x}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\large \int {cosecy}\ {\text{d}y}=\int sinx\ {\text{d}x}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\large \int \frac{1}{siny}\ {\text{d}y}=\int sinx\ {\text{d}x}\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>The differential equation is not separable</span></label> </p></div><div class="q-explanation"><p>We cannot separate sin(xy) into the form f(x)g(y)</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>Here is a solution to a differential equation. Is the work all correct? If not, which step is wrong?</p><p style="margin-left: 80px;"><span class="math-tex">\(\large \quad \quad\quad \quad \quad \quad \frac{\text{d}y}{\text{d}x}=\frac{x^2+1}{2y}\)</span></p><p style="margin-left: 80px;"><span class="math-tex">\(\large \text{step 1} \quad \quad \int {2y}\ {\text{d}y}=\int (x^2+1 )\ {\text{d}x} \)</span></p><p style="margin-left: 80px;"><span class="math-tex">\(\large \text{step 2} \quad \quad y^2=\frac{x^3}{3}+x+c \)</span></p><p style="margin-left: 80px;"><span class="math-tex">\(\large \text{step 3} \quad \quad y=\pm\sqrt{\frac{x^3}{3}+x+c}\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>step 2</span></label> </p><p><label class="radio"> <input type="radio"> <span>step 3</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>all correct</span></label> </p><p><label class="radio"> <input type="radio"> <span>step 1</span></label> </p></div><div class="q-explanation"><p>All step in this solution are correct!</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>Here is a solution to a differential equation. Is the work all correct? If not, which step is wrong?</p><p style="margin-left: 80px;"><span class="math-tex">\(\large \quad \quad\quad \quad \quad \quad \frac{\text{d}y}{\text{d}x}=\frac{lnx}{y}\)</span></p><p style="margin-left: 80px;"><span class="math-tex">\(\large \text{step 1} \quad \quad \int {y}\ {\text{d}y}=\int lnx\ {\text{d}x} \)</span></p><p style="margin-left: 80px;"><span class="math-tex">\(\large \text{step 2} \quad \quad y^2=xlnx-x+c \)</span></p><p style="margin-left: 80px;"><span class="math-tex">\(\large \text{step 3} \quad \quad y=\pm\sqrt{xlnx-x+c}\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>step 1</span></label> </p><p><label class="radio"> <input type="radio"> <span>all correct</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>step 2</span></label> </p><p><label class="radio"> <input type="radio"> <span>step 3</span></label> </p></div><div class="q-explanation"><p>The integration on the left-hand side in step 2 is incorrect</p><p style="margin-left: 80px;"><span class="math-tex">\(\int {y}\ {\text{d}y}=\frac{y^2}{\color{red}{\bf{2}}}+c\)</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>Here is a solution to a differential equation. Is the work all correct? If not, which step is wrong?</p><p style="margin-left: 80px;"><span class="math-tex">\(\large \quad \quad\quad \quad \quad \quad \frac{\text{d}y}{\text{d}x} = -xe^y\)</span></p><p style="margin-left: 80px;"><span class="math-tex">\(\large \text{step 1} \quad \quad \int {e^{-y}}\ {\text{d}y}=\int -x\ {\text{d}x} \)</span></p><p style="margin-left: 80px;"><span class="math-tex">\(\large \text{step 2} \quad \quad -e^{-y}=- \frac{x^2}{2}+c\)</span></p><p style="margin-left: 80px;"><span class="math-tex">\(\large \text{step 3} \quad \quad e^{-y}= \frac{x^2}{2}+c_1\)</span></p><p style="margin-left: 80px;"><span class="math-tex">\(\large \text{step 4} \quad \quad {-y}= ln({\frac{x^2}{2}+c_1)}\)</span></p><p style="margin-left: 80px;"><span class="math-tex">\(\large \text{step 5} \quad \quad {y}= ln({-\frac{x^2}{2}+c_2)}\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>step 4</span></label> </p><p><label class="radio"> <input type="radio"> <span>all correct</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>step 5</span></label> </p><p><label class="radio"> <input type="radio"> <span>step 2</span></label> </p></div><div class="q-explanation"><p>The final step should be</p><p><span class="math-tex">\(\large \text{step 5} \quad \quad {y}=- ln({\frac{x^2}{2}+c_1)}\)</span></p><p>We can simplify this further</p><p><span class="math-tex">\(\large \text{step 6} \quad \quad {y}= ln({\frac{x^2+c_2}{2})}^{-1}\)</span></p><p><span class="math-tex">\(\large \text{step 7} \quad \quad {y}= ln({\frac{2}{x^2+c_2})}\)</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>Here is a solution to a differential equation. Is the work all correct? If not, which step is wrong?</p><p style="margin-left: 120px;"><span class="math-tex">\(\quad \quad \quad \quad \large \frac{\text{d}y}{\text{d}x}=e^{x-y}\)</span></p><p style="margin-left: 120px;"><span class="math-tex">\(\large \text{step 1} \quad \quad \frac{\text{d}y}{\text{d}x}=e^{x} \cdot e^{-y}\)</span></p><p style="margin-left: 120px;"><span class="math-tex">\(\large \text{step 2} \quad \quad \int {e^{-y}}\ {\text{d}y}=\int e^{x}\ {\text{d}x} \)</span></p><p style="margin-left: 120px;"><span class="math-tex">\(\large \text{step 3} \quad \quad -e^{-y}=e^{x}+c\)</span></p><p style="margin-left: 120px;"><span class="math-tex">\(\large \text{step 4} \quad \quad e^{-y}=-e^x+c_1\)</span></p><p style="margin-left: 120px;"><span class="math-tex">\(\large \text{step 5} \quad \quad y=-ln(-e^x+c_1)\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>step 1</span></label> </p><p><label class="radio"> <input type="radio"> <span>all correct</span></label> </p><p><label class="radio"> <input type="radio"> <span>step 5</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>step 2</span></label> </p></div><div class="q-explanation"><p>It might to help of step 1 written as follows</p><p><span class="math-tex">\(\large \text{step 1} \quad \quad \frac{\text{d}y}{\text{d}x}=\frac{e^{x}}{e^y}\)</span></p><p>and so step 2 should be</p><p><span class="math-tex">\(\large \text{step 2} \quad \quad \int {e^{y}}\ {\text{d}y}=\int e^{x}\ {\text{d}x} \)</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>Which of the following is the correct solution to the following differential equation</p><p><span class="math-tex">\(\quad \quad \quad \quad \large \frac{\text{d}y}{\text{d}x}=y\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\large y=\pm\sqrt{x+C}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\large y=e^{x}+c\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\large y=c e^x\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\large y= e^{cx}\)</span></span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(\large \int \frac{1}{y}{\text{d}y}=\int 1 \ {\text{d}x}\)</span></p><p><span class="math-tex">\(\large ln|y|=x+c\)</span></p><p><span class="math-tex">\(\large y=e^{x+c}\)</span></p><p><span class="math-tex">\(\large y=c_1 e^x\)</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>Which of the following is the correct solution to the following differential equation</p><p><span class="math-tex">\(\quad \quad \quad \quad \large \frac{\text{d}y}{\text{d}x}=3x^2y\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\large y=e^{x^3}+c\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span>none of these answers are correct</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\large y=ce^{x^3}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\large y=e^{x^3}\)</span></span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(\large \int \frac{1}{y}{\text{d}y}=\int 3x^2 \ {\text{d}x}\)</span></p><p><span class="math-tex">\(\large ln|y|=x^3+c\)</span></p><p><span class="math-tex">\(\large y=e^{x^3+c}\)</span></p><p><span class="math-tex">\(\large y=c_1e^{x^3}\)</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>Which of the following is the correct solution to the following differential equation</p><p><span class="math-tex">\(\quad \quad \quad \quad \large \frac{\text{d}y}{\text{d}x}=\frac{x}{y}\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(y=cx\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\large y=\sqrt{x^2+2c}\)</span></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\large y=\pm \sqrt{x^2+c}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\large y=\sqrt{x^2+c}\)</span></span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(\large \int {y}\ {\text{d}y}=\int x\ {\text{d}x}\)</span></p><p><span class="math-tex">\(\large \frac{y^2}{2}= \frac{x^2}{2}+c\)</span></p><p><span class="math-tex">\(y^2=x^2+c_1\)</span></p><p><span class="math-tex">\(\large y=\pm\sqrt{x^2+c_1}\)</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>Which of the following is the correct solution to the following differential equation</p><p><span class="math-tex">\(\quad \quad \quad \quad \large \frac{\text{d}y}{\text{d}x}=\frac{sec^2x}{sec\ y}\)</span></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>none of the answers are correct</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(y=arcsin(tanx+c)\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span>this is not a separable differential equation</span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(y=arccos(tanx+c)\)</span></span></label> </p></div><div class="q-explanation"><p><span class="math-tex">\(\large \int {cosy}\ {\text{d}y}=\int sec^2x\ {\text{d}x}\)</span></p><p><span class="math-tex">\(\large siny = tanx+c\)</span></p><p><span class="math-tex">\(y=arcsin(tanx+c)\)</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> </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="867"> <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>Solve the differential equation given that y(0) = 2</p> <p><span class="math-tex">\({\large \frac{dy}{dx}=\frac{e^x}{y}, \quad y>0}\)</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>This is a separable differential equation.</p> <p><content>Re-arrange the equation, integrating the right-hand side with respect to y and the left-hand side with respect to x</content></p> <p><img alt="" src="../../files/integration/differential-equations/esq-de_sep1hint.png" style="width: 150px; height: 48px;"></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_des_sep1.jpg" style="width: 377px; height: 413px;"></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="894"> <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/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>a) Write <span class="math-tex">\({\huge \frac{1}{4-x^2}}\)</span>as the sum of two partial fractions</p> <p>b) Hence, given that y(0) = 0, find the particular solution of the differential equation in the form</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>b) This is a separable differential equation</p> <p><content><img alt="" src="../../files/integration/differential-equations/esq-de_sep2hint.png" style="width: 300px; height: 90px;"></content></p> <p>Use the result from part a) to re-write the second integrand</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_sep2a.png" style="width: 600px; height: 402px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_sep2b.png" style="width: 600px; height: 839px;"></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="899"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p>a) Express <span class="math-tex">\(\frac{3}{x(x-3)}\)</span> as the sum of two partial fractions</p> <p>The population of a species of fish can be modelled by the differential equation <span class="math-tex">\(\large \frac{\text{d}N}{\text{d}t}=\frac{2}{3}N(N-3)cos2t\)</span></p> <p>where <em><strong>N</strong></em> = population in thousands, <em><strong>t</strong></em> = time in years</p> <p>b) Given that initially the population of fish is 4000, show that <span class="math-tex">\(\large N=\frac{12}{4-e^{sin{2t}}}\)</span></p> <p>c) How many days during the first year is the population of fish above 8000?</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) Work out <span class="math-tex">\(\large \frac{3}{x(x-3)}\equiv\frac{A}{x}+\frac{B}{x-3}\)</span><content></content></p> <p>b) This is a variables separable differential equation</p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_sep3hinta.png" style="width: 300px; height: 49px;"></p> <p>Solve for <em><strong>N = 4 , t = 0</strong></em></p> <p>c) Use your graphical calculator to find two values in the first year where <em><strong>N > 8</strong></em></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_sep3a.png" style="width: 550px; height: 465px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_sep3b.png" style="width: 600px; height: 621px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_sep3c.png" style="width: 550px; height: 442px;"></p> <p><img alt="" src="../../files/integration/differential-equations/esq_de_sep3d.png" style="width: 580px; height: 534px;"></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 class="page-container panel-self-assessment" data-id="2107"> <div class="panel-heading">MY 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