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      <text>19.3.2.3 Finding the complementary function</text>
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      <text><![CDATA[<h4 class="HELM_example">Example</h4>
<p>Find the auxiliary equation of the differential equation: <span class="math inline">\[
a{\mathrm{d}^2y\over \mathrm{d}x^2}+b{\mathrm{d}y\over \mathrm{d}x}+cy=0\]</span></p>

<h4 class="HELM_solution">Solution</h4>

<p>We try a solution of the form <span class="math inline">\(y=\text{e}^{kx}\)</span> so that <span class="math inline">\[
{{\mathrm{d}y\over \mathrm{d}x}=k\text{e}^{kx}}
\qquad\text{and}\qquad 
{{\mathrm{d}^2y\over \mathrm{d}x^2}=k^2\text{e}^{kx}}.\]</span> Substitution into the given differential equation yields: <span class="math inline">\[ ak^2\text{e}^{kx}+bk\text{e}^{kx}+c\text{e}^{kx}=0\qquad\text{that is}\qquad (ak^2+bk+c)\text{e}^{kx}=0\]</span> Since
    this equation is to be satisfied for all values of <span class="math inline">\(x\)</span>, then <span class="math inline">\[ ak^2+bk+c=0\]</span> is the required auxiliary equation.</p>

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<div class="HELM_keypoint">
    <h4>Key Point</h4>
    <p>The <span><strong>auxiliary equation</strong></span> of <span class="math inline">\[{a{\mathrm{d}^2y\over \mathrm{d}x^2}+b{\mathrm{d}y\over \mathrm{d}x}+cy=0
\qquad\text{is}\qquad
ak^2+bk+c=0}\]</span> where <span class="math inline">\(y=e^{kx}\)</span></p>
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