The general solution of the differential equation $\frac{dy}{dx}= xy + x + y + 1$ is

$\log_e|y+1| = \frac{1}{2}x^2+x+C$, where C is constant of integration

The correct answer is Option (2) → $\log_e|y+1| = \frac{1}{2}x^2+x+C$, where C is constant of integration

Given:

$\frac{dy}{dx}=xy+x+y+1$

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

Separate variables:

$\frac{dy}{y+1}=(x+1)\,dx$

Integrate:

$\int \frac{dy}{y+1}=\int (x+1)\,dx$

$\log|y+1|=\frac{x^{2}}{2}+x+C$