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      <text>19.3.5.1 Finding the general solution of a snd order linear inhomogeneous ODE</text>
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      <text><![CDATA[<h4>Finding the general solution of a second order linear inhomogeneous ODE</h4>
<p>The general solution of a second order linear inhomogeneous equation is the sum of its particular integral and the complementary function. In previous subsections you learned how to find a complementary function, and just before you learnt how to find a particular integral. We now put these together to find the general solution.</p>

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<h4 class="HELM_example">Example</h4>
<p>Find the general solutions of</p>
\[\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + 3 \frac{\mathrm{d}y}{\mathrm{d}x} - 10y = 3x^2.\]
<h4 class="HELM_solution">Solution</h4>
<p>The complementary function was found in a previous example to be</p>
\[y_\text{cf} = Ae^{2x} + Be^{-5x}.\]
<p>The particular integral is found by trying a solution of the form \(y = ax^2 + bx + c\), so that</p>
\[\frac{\mathrm{d}y}{\mathrm{d}x} = 2ax + b \qquad \text{and} \qquad \frac{\mathrm{d}^2y}{\mathrm{d}x^2} 2a.\]
<p>Substituting into the differential equation gives:</p>
\[2a + 3(2ax + b) - 10(ax^2 + bx + c) = 3x^2.\]
<p>Comparing constants: &emsp; \(2a + 3b - 10c = 0\)</p>
<p>Comparing \(x\) terms: &emsp; \(6a - 10b = 0\)</p>
<p>Comparing \(x^2\) terms: &emsp; \(-10a = 3\)</p>
<p>So</p>
\[a = - \frac{3}{10}, \quad b = -\frac{9}{50}, \quad c = -\frac{57}{500},\]
<p>and</p>
\[y_\text{p}(x) = -\frac{3}{10}x^2 - \frac{9}{50}x - \frac{57}{500}.\]
<p>Thus, the general solution is</p>
\[y = y_\text{p}(x) + y_\text{cf}(x) = -\frac{3}{10}x^2 - \frac{9}{50}x - \frac{57}{500} + Ae^{2x} + Be^{-5x}.\]
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<h4>Key Point</h4>
<p>The general solution of a second order constant coefficient ordinary differential equation</p>
\[a\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + b \frac{\mathrm{d}y}{\mathrm{d}x} + cy = f(x)\]
<p>is</p>
\[y = y_\text{p} + y_\text{cf},\]
<p>being the sum of the particular integral and the complementary function.</p>
<p>\(y_\text{p}\) contains no arbitrary contants; \(y_\text{cf}\) contains two arbitrary constants.</p>
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