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Q.
The solution of the differential equation $x\left(y^{2} e^{x y}+e^{x / y}\right) d y=y\left(e^{x / y}-y^{2} e^{x y}\right) d x$ is
Differential Equations
Solution:
The given equation is
$ x y^{2} e^{x y} d y+x e^{x / y} d y =y e^{x / y} d x-y^{3} e^{x y} d x $
$\Rightarrow y^{2} \cdot e^{x y}(x d y+y d x) =e^{x / y}(y d x-x d y) $
$\Rightarrow e^{x y}(x d y+y d x) =e^{x / y}\left(\frac{y d x-x d y}{y^{2}}\right)$
$\Rightarrow e^{x y} \cdot d(x y) =e^{x / y}(d(x / y)) $
$\Rightarrow d\left(e^{x y}\right) =d\left(e^{x / y}\right)$
$ \Rightarrow e^{x y}=e^{x / y}+c $
or $x y =\ln \left(e^{x / y}+c\right)$