The general solution of the differential equation $\frac{dy}{dx}-\frac{y}{x}=x^2$ is :

$y=\frac{x^3}{2}+Cx$, where C is a constant

The correct answer is Option (2) → $y=\frac{x^3}{2}+Cx$, where C is a constant

$\frac{dy}{dx}-\frac{y}{x}=x^2$ → eq.

$I.F. = e^{-\int\frac{1}{x}dx}=e^{-\log x}=\frac{1}{x}$

multiplying eq with I.F.

$\int\frac{1}{x}\frac{dy}{dx}-\frac{y}{x}dx=\int xdx$

$⇒\frac{y}{x}=\frac{x^2}{2}+C$

$⇒y=\frac{x^3}{2}+Cx$