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213 lines
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213 lines
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<h1 class="center">Homogeneous Differential Equations</h1>
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<div class="def">
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<p>A <a href="differential-equations.html">Differential Equation</a> is an equation with a <a href="../sets/function.html">function</a> and one or more of its <a href="derivatives-introduction.html">derivatives</a>:</p>
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<p class="center"><img src="images/diff-eq-sep-var.svg" alt="differential equation dy/dx = 5xy"><br>
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Example: an equation with the function <b>y</b> and its
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derivative<b> <span class="intbl"> <em>dy</em> <strong>dx</strong> </span></b></p>
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</div>
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<p><span class="center">Here we look at a special method for solving "<a href="homogeneous-function.html">Homogeneous</a> Differential Equations"</span></p>
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<h2>Homogeneous Differential Equations</h2>
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<p>A first order <a href="differential-equations.html">Differential Equation</a> is <b>Homogeneous</b> when it can be in this form:</p>
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<p class="center larger"><span class="intbl"> <em>dy</em> <strong>dx</strong> </span> = F( <span class="intbl"> <em>y</em> <strong>x</strong> </span> )</p>
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<p>We can solve it using <a href="separation-variables.html">Separation of Variables</a> but first we create a new variable <b>v = <span class="intbl"> <em>y</em> x </span></b></p>
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<div class="so"> v = <span class="intbl"> <em>y</em> <strong>x</strong> </span> <i>which is also</i> y = vx </div>
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<div class="so"> And <span class="intbl"> <em>dy</em> <strong>dx</strong> </span> = <span class="intbl"> <em>d (vx)</em> <strong>dx</strong> </span> = v<span class="intbl"> <em>dx</em> <strong>dx</strong> </span> + x<span class="intbl"> <em>dv</em> <strong>dx</strong> </span> (by the <a href="derivatives-rules.html">Product Rule</a>) </div>
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<div class="so"> Which can be simplified to <span class="intbl"> <em>dy</em> <strong>dx</strong> </span> = v + x<span class="intbl"> <em>dv</em> <strong>dx</strong> </span> </div>
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<p>Using <b>y = vx</b> and<b> <span class="intbl"> <em>dy</em> <strong>dx</strong> </span> = v + x<span class="intbl"> <em>dv</em> <strong>dx</strong></span></b> we can solve the Differential Equation.</p>
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<p>An example will show how it is all done:</p>
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<div class="example">
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<h3>Example: Solve <span class="intbl"> <em>dy</em> <strong>dx</strong> </span> = <span class="intbl"><em>x<sup>2</sup> + y<sup>2</sup></em><strong>xy</strong></span></h3>
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<p>Can we get it in F( <span class="intbl"> <em>y</em> <strong>x</strong> </span> ) style?</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right"><span class="intbl"> <em>x<sup>2</sup> + y<sup>2</sup></em> <strong>xy</strong> </span></span></div>
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<div class="row"><span class="left">Separate terms:</span><span class="right"><span class="intbl"> <em>x<sup>2</sup></em> <strong>xy</strong> </span> + <span class="intbl"> <em>y<sup>2</sup></em> <strong>xy</strong> </span></span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right"><span class="intbl"> <em>x</em> <strong>y</strong> </span> + <span class="intbl"> <em>y</em> <strong>x</strong> </span></span></div>
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<div class="row"><span class="left">Reciprocal of first term:</span><span class="right">( <span class="intbl"> <em>y</em> <strong>x</strong> </span> )<sup>-1</sup> + <span class="intbl"> <em>y</em> <strong>x</strong> </span></span></div>
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</div>
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<p>Yes, we have a function of <span class="intbl"><em>y</em><strong>x</strong></span>.</p>
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<p>So let's go:</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right"><span class="intbl"> <em>dy</em> <strong>dx</strong> </span> = ( <span class="intbl"> <em>y</em> <strong>x</strong> </span> )<sup>-1</sup> + <span class="intbl"> <em>y</em> <strong>x</strong> </span></span></div>
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<div class="row"><span class="left"><b>y = vx</b> and<b> <span class="intbl"> <em>dy</em><strong>dx</strong> </span> = v + x<span class="intbl"> <em>dv</em><strong>dx</strong></span></b>:</span><span class="right">v + x<span class="intbl"> <em>dv</em> <strong>dx</strong> </span> = v<sup>-1</sup> + v</span></div>
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<div class="row"><span class="left">Subtract v from both sides:</span><span class="right">x<span class="intbl"> <em>dv</em> <strong>dx</strong> </span> = v<sup>-1</sup></span></div>
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</div>
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<p>Now use <a href="separation-variables.html">Separation of Variables</a>:</p>
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<div class="tbl">
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<div class="row"><span class="left">Separate the variables:</span><span class="right">v dv = <span class="intbl"> <em>1</em> <strong>x</strong> </span> dx</span></div>
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<div class="row"><span class="left">Put the integral sign in front:</span><span class="right"><span class="integral">∫</span>v dv = <span class="integral">∫</span><span class="intbl"> <em>1</em> <strong>x</strong> </span> dx</span></div>
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<div class="row"><span class="left">Integrate:</span><span class="right"><span class="intbl"> <em>v<sup>2</sup></em> <strong>2</strong> </span> = ln(x) + C</span></div>
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<div class="row"><span class="left">Then we make <b>C = ln(k)</b>:</span><span class="right"><span class="intbl"> <em>v<sup>2</sup></em> <strong>2</strong> </span> = ln(x) + ln(k)</span></div>
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<div class="row"><span class="left">Combine ln:</span><span class="right"><span class="intbl"> <em>v<sup>2</sup></em> <strong>2</strong> </span> = ln(kx)</span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right">v = ±√(2 ln(kx))</span></div>
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</div>
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<p>Now substitute back v = <span class="intbl"> <em>y</em> <strong>x</strong> </span></p>
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<div class="tbl">
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<div class="row"><span class="left">Substitute v = <span class="intbl"> <em>y</em> <strong>x</strong> </span>:</span><span class="right"><span class="intbl"> <em>y</em> <strong>x</strong> </span> = ±√(2 ln(kx))</span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right">y = ±x √(2 ln(kx))</span></div>
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</div>
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<p>And we have the solution.</p>
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<p>The positive portion looks like this:</p>
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<p class="center"><img src="images/x-sqrt-2lnkx.svg" alt="y = x sqrt(2 ln(kx))"></p>
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</div>
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<p> </p>
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<p>Another example:</p>
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<div class="example">
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<h3>Example: Solve <span class="intbl"> <em>dy</em> <strong>dx</strong> </span> = <span class="intbl"> <em>y(x−y)</em> <strong>x<sup>2</sup></strong> </span></h3>
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<p>Can we get it in F( <span class="intbl"> <em>y</em> <strong>x</strong> </span> ) style?</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right"><span class="intbl"> <em>y(x−y)</em> <strong>x<sup>2</sup></strong> </span></span></div>
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<div class="row"><span class="left">Separate terms:</span><span class="right"><span class="intbl"> <em>xy</em> <strong>x<sup>2</sup></strong> </span> − <span class="intbl"> <em>y<sup>2</sup></em> <strong>x<sup>2</sup></strong> </span></span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right"><span class="intbl"> <em>y</em> <strong>x</strong> </span> − ( <span class="intbl"> <em>y</em> <strong>x</strong> </span> )<sup>2</sup></span></div>
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</div>
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<p>Yes! So let's go:</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right"><span class="intbl"> <em>dy</em> <strong>dx</strong> </span> = <span class="intbl"> <em>y</em> <strong>x</strong> </span> − ( <span class="intbl"> <em>y</em> <strong>x</strong> </span> )<sup>2</sup></span></div>
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<div class="row"><span class="left"><b>y = vx</b> and <span class="intbl"> <em><b>dy</b></em> <b> <strong>dx</strong></b></span><b> = v + x<span class="intbl"> <em>dv</em><strong>dx</strong> </span></b></span><span class="right">v + x<span class="intbl"> <em>dv</em> <strong>dx</strong> </span> = v − v<sup>2</sup></span></div>
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<div class="row"><span class="left">Subtract v from both sides:</span><span class="right">x<span class="intbl"> <em>dv</em> <strong>dx</strong> </span> = −v<sup>2</sup></span></div>
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</div>
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<p>Now use <a href="separation-variables.html">Separation of Variables</a>:</p>
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<div class="tbl">
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<div class="row"><span class="left">Separate the variables:</span><span class="right">−<span class="intbl"> <em>1</em> <strong>v<sup>2</sup></strong> </span> dv = <span class="intbl"> <em>1</em> <strong>x</strong> </span> dx</span></div>
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<div class="row"><span class="left">Put the integral sign in front:</span><span class="right"><span class="integral">∫</span>−<span class="intbl"> <em>1</em> <strong>v<sup>2</sup></strong> </span> dv = <span class="integral">∫</span><span class="intbl"> <em>1</em> <strong>x</strong> </span> dx</span></div>
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<div class="row"><span class="left">Integrate:</span><span class="right"><span class="intbl"> <em>1</em> <strong>v</strong> </span> = ln(x) + C</span></div>
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<div class="row"><span class="left">Then we make <b>C = ln(k)</b>:</span><span class="right"><span class="intbl"> <em>1</em> <strong>v</strong> </span> = ln(x) + ln(k)</span></div>
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<div class="row"><span class="left">Combine ln:</span><span class="right"><span class="intbl"> <em>1</em> <strong>v</strong> </span> = ln(kx)</span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right">v = <span class="intbl"> <em>1</em> <strong>ln(kx)</strong> </span></span></div>
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</div>
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<p>Now substitute back v = <span class="intbl"> <em>y</em> <strong>x</strong> </span></p>
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<div class="tbl">
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<div class="row"><span class="left">Substitute v = <span class="intbl"> <em>y</em> <strong>x</strong> </span>:</span><span class="right"><span class="intbl"> <em>y</em> <strong>x</strong> </span> = <span class="intbl"> <em>1</em> <strong>ln(kx)</strong> </span></span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right">y = <span class="intbl"> <em>x</em> <strong>ln(kx)</strong> </span></span></div>
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</div>
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<p>And we have the solution.</p>
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<p>Here are some sample k values:</p>
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<p class="center"><img src="images/diff-eq-hom-2.svg" alt="y = x / ln(kx)"></p>
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</div>
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<p>And one last example:</p>
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<div class="example">
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<h3>Example: Solve <span class="intbl"> <em>dy</em> <strong>dx</strong> </span> = <span class="intbl"> <em>x−y</em> <strong>x+y</strong> </span></h3>
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<p>Can we get it in F( <span class="intbl"> <em>y</em> <strong>x</strong> </span> ) style?</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right"><span class="intbl"> <em>x−y</em> <strong>x+y</strong> </span></span></div>
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<div class="row"><span class="left">Divide through by x:</span><span class="right"><span class="intbl"> <em>x/x−y/x</em> <strong>x/x+y/x</strong> </span></span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right"><span class="intbl"> <em>1−y/x</em> <strong>1+y/x</strong> </span></span></div>
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</div>
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<p>Yes! So let's go:</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right"><span class="intbl"> <em>dy</em> <strong>dx</strong> </span> = <span class="intbl"> <em>1−y/x</em> <strong>1+y/x</strong> </span></span></div>
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<div class="row"><span class="left"><b>y = vx</b> and <span class="intbl"> <em><b>dy</b></em> <b> <strong>dx</strong></b></span><b> = v + x<span class="intbl"> <em>dv</em><strong>dx</strong> </span></b></span><span class="right">v + x<span class="intbl"> <em>dv</em> <strong>dx</strong> </span> = <span class="intbl"> <em>1−v</em> <strong>1+v</strong> </span></span></div>
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<div class="row"><span class="left">Subtract v from both sides:</span><span class="right">x<span class="intbl"> <em>dv</em> <strong>dx</strong> </span> = <span class="intbl"> <em>1−v</em> <strong>1+v</strong> </span> − v</span></div>
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<div class="row"><span class="left">Then:</span><span class="right">x<span class="intbl"> <em>dv</em> <strong>dx</strong> </span> = <span class="intbl"> <em>1−v</em> <strong>1+v</strong> </span> − <span class="intbl"> <em>v+v<sup>2</sup></em> <strong>1+v</strong> </span></span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right">x<span class="intbl"> <em>dv</em> <strong>dx</strong> </span> = <span class="intbl"> <em>1−2v−v<sup>2</sup></em> <strong>1+v</strong> </span></span></div>
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</div>
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<p>Now use <a href="separation-variables.html">Separation of Variables</a>:</p>
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<div class="tbl">
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<div class="row"><span class="left">Separate the variables:</span><span class="right"><span class="intbl"> <em>1+v</em> <strong>1−2v−v<sup>2</sup></strong> </span> dv = <span class="intbl"> <em>1</em> <strong>x</strong> </span> dx</span></div>
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<div class="row"><span class="left">Put the integral sign in front:</span><span class="right"><span class="integral">∫</span><span class="intbl"> <em>1+v</em> <strong>1−2v−v<sup>2</sup></strong> </span> dv = <span class="integral">∫</span><span class="intbl"> <em>1</em> <strong>x</strong> </span> dx</span></div>
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<div class="row"><span class="left">Integrate:</span><span class="right">−<span class="intbl"> <em>1</em> <strong>2</strong> </span> ln(1−2v−v<sup>2</sup>) = ln(x) + C</span></div>
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<div class="row"><span class="left">Then we make <b>C = ln(k)</b>:</span><span class="right">−<span class="intbl"> <em>1</em> <strong>2</strong> </span> ln(1−2v−v<sup>2</sup>) = ln(x) + ln(k)</span></div>
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<div class="row"><span class="left">Combine ln:</span><span class="right">(1−2v−v<sup>2</sup>)<sup>-½</sup> = kx</span></div>
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<div class="row"><span class="left">Square and Reciprocal:</span><span class="right">1−2v−v<sup>2</sup> = <span class="intbl"> <em>1</em> <strong>k<sup>2</sup>x<sup>2</sup></strong> </span></span></div>
|
||
</div>
|
||
<p>Now substitute back v = <span class="intbl"> <em>y</em> <strong>x</strong> </span></p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Substitute v = <span class="intbl"> <em>y</em> <strong>x</strong> </span>:</span><span class="right">1−2( <span class="intbl"> <em>y</em> <strong>x</strong> </span> )−( <span class="intbl"> <em>y</em> <strong>x</strong> </span> )<sup>2</sup> = <span class="intbl"> <em>1</em> <strong>k<sup>2</sup>x<sup>2</sup></strong> </span></span></div>
|
||
<div class="row"><span class="left">Multiply through by <b>x<sup>2</sup></b>:</span><span class="right">x<sup>2</sup>−2xy−y<sup>2</sup> = <span class="intbl"> <em>1</em> <strong>k<sup>2</sup></strong> </span></span></div>
|
||
</div>
|
||
<p>We are nearly there ... it is nice to separate out y though!<br>
|
||
We can try to factor <span class="larger">x<sup>2</sup>−2xy−y<sup>2</sup></span> but we must do some rearranging first:</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Change signs:</span><span class="right">y<sup>2</sup>+2xy−x<sup>2</sup> = − <span class="intbl"> <em>1</em> <strong>k<sup>2</sup></strong> </span></span></div>
|
||
<div class="row"><span class="left">Replace − <span class="intbl"> <em>1</em> <strong>k<sup>2</sup></strong> </span> by c:</span><span class="right">y<sup>2</sup>+2xy−x<sup>2</sup> = c</span></div>
|
||
<div class="row"><span class="left">Add 2x<sup>2</sup> to both sides:</span><span class="right">y<sup>2</sup>+2xy+x<sup>2 </sup>= 2x<sup>2</sup>+c</span></div>
|
||
<div class="row"><span class="left">Factor:</span><span class="right">(y+x)<sup>2</sup> = 2x<sup>2</sup>+c<span class="intbl"></span></span></div>
|
||
<div class="row"><span class="left">Square root:</span><span class="right">y+x = ±√(2x<sup>2</sup>+c)<span class="intbl"></span></span></div>
|
||
<div class="row"><span class="left">Subtract x from both sides:</span><span class="right">y = ±√(2x<sup>2</sup>+c)<span class="intbl"></span> − x<span class="intbl"></span></span></div>
|
||
</div>
|
||
<p>And we have the solution.</p>
|
||
<p>The positive portion looks like this:</p>
|
||
<p class="center"><img src="images/diff-eq-hom-3.svg" alt="y = sqrt(2x^2+c) - x"></p>
|
||
</div>
|
||
<p> </p>
|
||
|
||
<div class="questions">
|
||
<script>getQ(9419, 9420, 9421, 9422, 9423, 9424, 9425, 9426, 9427, 9428);</script>
|
||
</div>
|
||
|
||
<div class="related">
|
||
<a href="homogeneous-function.html">Homogeneous Functions</a>
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<a href="differential-equations.html">Differential Equation</a>
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<a href="index.html">Calculus Index</a>
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