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Solving Homogeneous Differential Equations

Solve xdy-y...

Question

Solve $x d y - y d x = x^{3} d y + x^{2} y d x$

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Solution

$x d y - y d x = x^{3} d y + x^{2} y d x$
$\Rightarrow \frac{x d y - y d x}{x^{2}} = \frac{x^{3} d y + x^{2} y d x}{x^{2}}$
$\int d (\frac{y}{x}) = x d y + y d x = \int d (x y)$
$∴ \frac{y}{x} - x y + C$

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