If $ (2+sin\, x) \frac{dy}{dx} + (y + 1) cos\, x = 0 $ and $y(0)  = 1$, then $y \left(\frac{\pi }{2}\right)$ is equal to

$\frac{1}{3}$

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

The given differential equation is

$(2+sin\, x) \frac{dy}{dx} + ( y+1) cos\, x= 0 $

$⇒(2+ sin x) dy + (y +1) cos\, x dx=0$

$⇒\frac{1}{y+1}dy + \frac{cosx}{2+sin x} dx=0$

Integrating both sides, we get

$log (y +1) + log (2+ sin x ) = log C$

$⇒(y+1) (2+ sinx ) = C$ ...........(i)

It is given that $y(0) = 1 $ i.e. $y = 1 $ when $x=0$. Putting $x=0, y = 1 $ in (i), we get

$2(2+0) =C⇒C=4$

Putting $C=4$ in (i), we obtain

$(y+1) (2+sin x) = 4$ ...........(ii)

Putting $x=\frac{\pi }{2}$ in (ii), we get $y=\frac{1}{3}$.