Question

Solve

$\frac{dy}{dx}=x^{2}y+y$

Answer 1

$\frac{dy}{dx}=x^{2}y+y$

$\Rightarrow \frac{dy}{dx}=(x^2+1)y$

$\Rightarrow \frac{dy}{y}=(x^2+1)dx$

Integrating on both sides we get

$logy=\frac{x^3}{3}+x+c$

$\Rightarrow 3logy=x^3+3x+3c$

$\Rightarrow log{y}^3=x^3+3x+c_1$

where $c_1=3c$.