<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="description">
    <name>
      <text>25.1.1.3 Second order, homogeneous, constant coeff ODES</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[We shall also need to be familiar with solutions to second order, homogeneous, constant coefficient ODEs, summarised in the Key Point below.

<div class="HELM_keypoint">
<h4>Key Point</h4>
<p> A second order ODE of the form
<table width="100%">
<tr><td>\[a\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}+b\dfrac{\mathrm{d}y}{\mathrm{d}x}+cy=0\]</td><td>\((1)\)</td></tr>
</table>
where \(a,b,c\) are constants, has an auxiliary equation
<table width="100%">
<tr><td>\[am^2+bm+c=0\]</td><td >\((2)\)</td></tr>
</table>
obtained by inserting the trial solution \(y=e^{mx}\) in \((1)\).
</p>
<p>The general solution of \((1)\) then depends on the solutions (or roots) of the quadratic equation \((2)\).</p>
<ol type="a">
<li><p>If \((2)\) has real, distinct roots \(m=m_1\) and \(m=m_2\) then \[y=Ae^{m_1x}+Be^{m_2x}\]</p></li>
<li><p>If \((2)\) has a repeated root \(m=m_1\) then \[y=(A+Bx)e^{m_1x}\]</p></li>
<li><p>If \((2)\) has complex roots (which will be a conjugate pair) \(m=\alpha\pm j \beta\) then \[ y=e^{\alpha x}(A\cos \beta x+B\sin \beta x)\]</p></li>
</ol>
<p> Note that in each of these cases (a) to (c) the general solution is a linear combination of two particular solutions:
<ul>
<li>For (a) they are \(e^{m_1x}\) and \(e^{m_2x}\).</li>
<li>For (b) they are \(e^{m_1x}\) and \(xe^{m_1x}\).</li>
<li>For (c) they are \(e^{\alpha x}\cos\beta x\) and \(e^{\alpha x}\sin \beta x\).</li>
</ul>
</p>
</div>]]></text>
    </questiontext>
    <generalfeedback format="moodle_auto_format">
      <text/>
    </generalfeedback>
    <defaultgrade>0</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
  </question>
</quiz>
