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What is the solution of the differential equation $x d y-y d x=x y^{2} d x ? \quad$
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The correct answer is:
$y x^{2}+2 x=2 c y$
Given, $x d y-y d x=x y^{2} d x$
$\Rightarrow \frac{x d y-y d x}{y^{2}}=x d x \Rightarrow \frac{y d x-x d y}{y^{2}}=-x d x$
$\Rightarrow \int d\left(\frac{x}{y}\right)=-\int x d x \Rightarrow \frac{x}{y}=\frac{-x^{2}}{2}+c=\frac{-x^{2}+2 c}{2}$
$\Rightarrow \quad y x^{2}+2 x=2 c y$
$\Rightarrow \frac{x d y-y d x}{y^{2}}=x d x \Rightarrow \frac{y d x-x d y}{y^{2}}=-x d x$
$\Rightarrow \int d\left(\frac{x}{y}\right)=-\int x d x \Rightarrow \frac{x}{y}=\frac{-x^{2}}{2}+c=\frac{-x^{2}+2 c}{2}$
$\Rightarrow \quad y x^{2}+2 x=2 c y$
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