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      <text>19.1.2.1 The general solution of a differential equation</text>
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      <text><![CDATA[<h4>The general solution of a differential equation</h4>
<p>Consider the equation <span class="math inline">\(y=A \text{e}^{2x}\)</span> where <span class="math inline">\(A\)</span> is an arbitrary constant. If we differentiate it we obtain <span class="math inline">\[
{dy\over dx}=2A\text{e}^{2x}\]</span> and so, since <span class="math inline">\[y=A\text{e}^{2x}\]</span> we obtain <span class="math inline">\[ {dy\over {dx}}=2y.\]</span> Thus a differential equation satisfied by <span class="math inline">\(y\)</span> is <span class="math inline">\[{dy\over dx}=2y.\]</span> Note that we have eliminated the arbitrary constant.</p>
<p>Now consider the equation <span class="math inline">\[y=A\cos 3x+B\sin 3x\]</span> where <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span> are arbitrary constants. Differentiating, we obtain <span class="math inline">\[ 
{dy\over dx}=-3A\sin 3x+3B\cos 3x.\]</span></p>
<p>Differentiating a second time gives <span class="math inline">\[
{d^2y\over dx^2}=-9\,A\cos 3x-9\,B\sin 3x.\]</span> The right-hand side is simply <span class="math inline">\((-9)\)</span> times the expression for <span class="math inline">\(y\)</span>. Hence <span class="math inline">\(y\)</span> satisfies the differential
    equation <span class="math inline">\[ 
{d^2y\over dx^2}=-9y.\]</span></p>]]></text>
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