<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="description">
    <name>
      <text>25.3.2.3  Method of separation of variables - general approach</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<b>Wave equation</b>
<table width="100%">
    <tbody>
        <tr>
            <td>\(\qquad\displaystyle \dfrac{\partial^2u}{\partial x^2}=\dfrac{1}{c^2}\dfrac{\partial^2u}{\partial t^2}\)
            </td>
            <td>\((6)\)</td>
        </tr>
    </tbody>
</table>


<hr>
<h4 class="HELM_example">Example</h4>
<p>By following a similar procedure to the above, assume a product solution </p>
<p><span class="math inline">\(\qquad\displaystyle
  u(x,t)=X(x)T(t)\)</span> </p>
for the wave equation and find the two ODEs satisfied by <span class="math inline">\(X(x)\)</span> and <span class="math inline">\(T(t)\)</span>.
<p></p>
<ol class="HELM_parts">
    <li>First obtain <span class="math inline">\(\dfrac{\partial^2u}{\partial
x^2}\)</span> and <span class="math inline">\(\dfrac{\partial^2u}{\partial
t^2}\)</span>:</li>
    <li>Now substitute these results into \((6)\) and transpose so the variables are separated i.e. all functions of <span class="math inline">\(x\)</span> are on the left-hand side, all funtions of <span class="math inline">\(t\)</span> on the right-hand
        side:
    </li>
    <li>Finally, write down the required ordinary differential equations:</li>
</ol>


<h4 class="HELM_solution">Solution</h4>
<p></p>
<ol class="HELM_parts">
    <li>\( u=X(x)T(t)\) gives \(\dfrac{\partial^2u}{\partial x^2}=X''T\) and \(\dfrac{\partial^2u}{\partial t^2}=XT''\)
    </li>
    <li>We get \(X''T= \dfrac{1}{c^2}XT''\) and, transposing, \(\dfrac{X''}{X}=\dfrac{1}{c^2} \dfrac{T''}{T}\)</li>
    <li>Equating both sides to the same constant \(K\) gives
        <p>\(\qquad \dfrac{X''}{X}=K\) or \(\dfrac{d^2X}{dx^2}-KX=0\)</p>
        <p>and</p>
        \(\qquad \dfrac{1}{c^2} \dfrac{T''}{T}=K \qquad\) or \(\qquad \dfrac{d^2 T}{dt^2}-Kc^2T=0\)</li>
</ol>
<hr> The solution of the ODEs \((7)\) and \((8)\) has been obtained earlier, and will depend on the sign of \(K\).]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text/>
    </generalfeedback>
    <defaultgrade>0</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
  </question>
</quiz>
