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      <text>19.2.6.1 The Integrating factor</text>
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      <text><![CDATA[<h4>The Integrating factor</h4>
<p>The equation\[x^2{\mathrm{d}y\over \mathrm{d}x}+3x\ y=x^3\] is not exact.  However, if we multiply it by \(x\) we obtain the equation \[ x^3{\mathrm{d}y\over \mathrm{d}x}+3x^2y=x^4.\]
This can be re-written as \[ {\mathrm{d}\over \mathrm{d}x} (x^3y)=x^4\] which is an exact equation with solution
\[ x^3y=\int x^4 \,\mathrm{d}x\]
so \[x^3y={1\over 5}x^5+C\] and hence
\[ y={1\over 5}x^2+{C\over x^3}.\]
The function by which we multiplied the given differential equation in order to make it exact is called an <strong> integrating factor</strong>. In this example the integrating factor is simply \(x\).</p>]]></text>
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