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

JEE Main · 2017 · 02 Apr 2017 · Offline · Differential Equations

(2 + sin x) \frac{d y}{d x} + (y + 1) cos x = 0

and

y (0) = 1

, then

y (\frac{π}{2})

is equal to :

Answer: (B)

The equation separates to

\frac{d y}{y + 1} = - \frac{cos x}{2 + sin x} d x

. Integrating both sides,

ln ∣ y + 1∣ = - ln (2 + sin x) + C

, so

(y + 1) (2 + sin x)

stays constant. The start

y (0) = 1

gives

(2) (2) = 4

, hence

y + 1 = \frac{4}{2 + sin x}

. At

x = \frac{π}{2}

y + 1 = \frac{4}{3}

, so

y = \frac{1}{3}

JEE Main · 2017 · 02 Apr 2017 · Offline · Single correct option

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