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      <text>25.1.1.6 General solutions of two ODEs</text>
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      <text><![CDATA[<p>The two Exercises above can be generalised as in the following Key Point.</p>
<div class="HELM_keypoint">
<h4>Key Point</h4>
<ol>
<li>The general solution to: \(\dfrac{d^2y}{dx^2}-n^2y=0\) is \[ y=Ae^{nx}+B e^{-nx}\] or, equivalently using hyperbolic functions, \[ y=C\cosh nx+D\sinh nx.\]</li>
<li>The general solution to: \(\dfrac{d^2y}{dx^2}+n^2y=0\) is \[ y=A\cos nx+B\sin nx.\]</li>
</ol>
</div>
<br>
<p>Those of you who are familiar with elementary dynamics will recognise the second
differential equation in the above Key Point as modelling <strong> simple harmonic motion</strong>.</p>]]></text>
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