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      <text>19.2.5.1 Recognising an exact equation</text>
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      <text><![CDATA[<h4>Recognising an exact equation</h4>
<p>The equation \(\dfrac{\mathrm{d}}{\mathrm{d}x}(yx)=3x^2\) is exact, as we have seen. If we expand the left-hand side of this equation (i.e. differentiate the product) we obtain
\[ x \dfrac{\mathrm{d}y}{\mathrm{d}x}+y.\]
Hence the equation \[ x \dfrac{\mathrm{d}y}{\mathrm{d}x}+y=3x^2\] must be exact, but it is not so obvious that it is exact as in the original form. This leads to the following Key
Point:</p>
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<h4>Key Point</h4>
<p>The equation
\[ f(x){\mathrm{d}y\over \mathrm{d}x}+y\ f'(x)=g(x)\] is exact.  It can be re-written as
\[\dfrac{\mathrm{d}}{\mathrm{d}x}(y\, f(x))=g(x) \quad\text{so that}\quad  y\, f(x)=\int g(x)\, \mathrm{d}x .\]</p>
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