Find the general solution of $x d y + y d x = x^{2} y d y$

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Solution

$x d y + y d x = x^{2} y d y$

$\Rightarrow x d x + y d x = (x y)^{2} \frac{d y}{y}$

$\Rightarrow \frac{d (x y)}{(x y)^{2}} = \frac{d y}{y}$ $(∵ d (x y) = x d y + y d x)$

$\Rightarrow \frac{(x y)^{- 2 + 1}}{(- 2 + 1)} = ln | y | + ln | c |$ $c = c o n s t a n t$

$\Rightarrow \frac{- 1}{x y} = ln | c y |$

$\Rightarrow y = k e^{- 1 / x y}$ $k = c o n s t a n t$

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