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    <name>
      <text>19.3.4.1 Finding a particular integral</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<h4>Finding a particular integral</h4>
<p>In the previous subsection we explained what is meant by a particular integral. Now we look at a
    simple method to find a particular integral. In fact our method is rather crude. It involves trial and
    error and educated guesswork. We try solutions which are of the same general form as the \(f(x)\) on
    the right-hand side.</p>

<hr>
<h4 class="HELM_example">Example</h4>
<p>Find a particular integral of the equation</p>
\[\frac{\mathrm{d}^2y}{\mathrm{d}x^2} - \frac{\mathrm{d}y}{\mathrm{d}x} - 6y = e^{2x}.\]
<h4 class="HELM_solution">Solution</h4>
<p>We shall attempt to find a solution of the inhomogeneous problem by trying a function of the same
    form as that on the right-hand side of the ODE. In particular, let us try \(y(x) = Ae^{2x}\),
    where A is a constant that we shall now determine. If \(y(x) = Ae^{2x}\) then
    </p>
\[\frac{\mathrm{d}y}{\mathrm{d}x} = 2Ae^{2x} \qquad \text{and} \qquad
\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = 4Ae^{2x}.\]
<p>Subsitution in the ODE gives:</p>
\[4Ae^{2x} - 2Ae^{2x} - 6Ae^{2x} = e^{2x},\]
<p>that is,</p>
\[-4Ae^{2x} = e^{2x}.\]
<p>To ensure that \(y\) is a solution, we require \(-4A = 1\), that is, \(\displaystyle A = -\frac{1}{4}\).</p>
<p>Therefore, the particular integral is \(\displaystyle y_\text{p} = -\frac{1}{4}e^{2x}\).</p>
<hr>

<p>In the previous example we chose a trial solution \(Ae^{2x}\) of the same form as the ODE's right-hand
    side. The following table provides a summary of the trial solutions which should be tried for various forms
    of the right-hand side.
</p>
<table class="HELM_table">
    <thead>
        <tr>
            <th>\(f(x)\)</th>
            <th>Trial solution</th>
        </tr>
    </thead>
    <tbody>
        <tr>
            <td>constant term: \(c\)</td>
            <td>constant term: \(k\)</td>
        </tr>
        <tr>
            <td>linear: \(ax + b\)</td>
            <td>linear: \(Ax + B\)</td>
        </tr>
        <tr>
            <td>polynomial in \(x\) of degree \(r\): \(ax^r + \cdots + bx + c\)</td>
            <td>polynomial in \(x\) of degree \(r\): \(Ax^r + \cdots + Bx + C\)</td>
        </tr>
        <tr>
            <td>\(a \cos kx\)</td>
            <td>\(A \cos kx + B \sin kx\)</td>
        </tr>
        <tr>
            <td>\(a \sin kx\)</td>
            <td>\(A \cos kx + B \sin kx\)</td>
        </tr>
        <tr>
            <td>\(ae^{kx}\)</td>
            <td>\(Ae^{kx}\)</td>
        </tr>
        <tr>
            <td>\(ae^{-kx}\)</td>
            <td>\(Ae^{-kx}\)</td>
        </tr>
    </tbody>
</table>]]></text>
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