Question

$xdy+(y-x^3)dx=0$

Answer 1

By seperating the terms $xdy+ydx=x^{3}dx$

We know that $\frac{d\left(xy\right)}{dx}=x\frac{dy}{dx}+y\frac{dx}{dx}$

Now, $xdy+ydx=d\left(xy\right)$

$\Rightarrow xdy+ydx=x^{3}dx$

$\Rightarrow d\left(xy\right)dx=x^{3}dx$

$\Rightarrow \int d\left(xy\right)dx=\int x^{3}dx$

$\Rightarrow xy=\frac{x^4}{4}+c$ where $c$ is the constant of integration

$\Rightarrow y=\frac{x^3}{4}+cx$ is the solution of the above equation