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      <text>19.2.5.2 Solve an exact equation</text>
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      <text><![CDATA[<h4 class="HELM_example">Example</h4>
<p> Solve the equation \[ x^3 \dfrac{\mathrm{d}y}{\mathrm{d}x}+3x^2y=x\]</p>
<h4 class="HELM_solution">Solution</h4>
<p>Comparing this equation with the form in the Key Point before we see that \(f(x)=x^3\) and \(g(x)=x\). Hence the equation can be written \[ \dfrac{\mathrm{d}}{\mathrm{d}x}(yx^3)=x,\] which has solution \[yx^3=\int x\, \mathrm{d}x=\dfrac{1}{2}x^2+C.\]Therefore \[ y=\dfrac{1}{2x}+\dfrac{C}{x^3}.\]
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