Consider the differential equation $\frac{dy}{dx}=\frac{y+1}{x+1}$, and y = 0 where x = 2. The value of y at x = 3 is:

$\frac{1}{3}$

The correct answer is option (2) → $\frac{1}{3}$

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

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

$⇒ln|y+1|=ln|x+1|+C$

and,

$y=0$ when $x=2$

$∴ln|1|=ln|3|+C$

$⇒C=-ln|3|$

$∴ln|y+1|=ln|4|-ln|3|$

$ln|y+1|=ln|\frac{4}{3}|$

$⇒y+1=\frac{4}{3}$

$⇒3y=1$

$⇒y=\frac{1}{3}$