Find the general solution of the differential equation $\frac{dy}{dx} = \frac{x+1}{2-y}, (y \neq 2)$.

$x^2 + y^2 + 2x - 4y + C = 0$

The correct answer is Option (2) → $x^2 + y^2 + 2x - 4y + C = 0$ ##

We have

$\frac{dy}{dx} = \frac{x+1}{2-y} \quad \dots(1)$

Separating the variables in equation (1), we get

$(2-y) \, dy = (x+1) \, dx \quad \dots(2)$

Integrating both sides of equation (2), we get

$\int (2-y) \, dy = \int (x+1) \, dx$

$\text{or } 2y - \frac{y^2}{2} = \frac{x^2}{2} + x + C_1$

$\text{or } x^2 + y^2 + 2x - 4y + 2C_1 = 0$

$\text{or } x^2 + y^2 + 2x - 4y + C = 0, \text{ where } C = 2C_1$

which is the general solution of equation (1).