Question

The solution of the differential equation $\left(2 x-10 y^{3}\right) \frac{d y}{d x}+y=0$ is $x y^{n}=2 y^{m}+C$ then find $n+m$.

Answer 1

$y \frac{d x}{d y}+2 x=10 y^{3}$
$\Rightarrow \frac{d x}{d y}+\frac{2 x}{y}=10 y^{2}$
$I.F. =e^{\int \frac{2}{y} d y}=y^{2}$
$x . y^{2}=\int 10 y^{4} d y$
$\Rightarrow x y^{2}=2 y^{5}+c$