Question

If $y=y\left(x\right)$ is the solution of the differential equation $\left(\frac{2+\sin x}{y+1}\right)\frac{dy}{dx}+\cos x=0$ with $y\left(0\right)=1$, then $y\left(\frac{\pi }{2}\right)$ __.

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $1$
• D. $\frac{4}{3}$

Answer 1

The correct option is A $\frac{1}{3}$

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

$\Rightarrow \frac{dy}{y+1}=-\frac{\cos x}{2+\sin x}dx$, Rearrange

$\Rightarrow \left(\ln \right(y+1\left)\right)=-\left(\ln \right(2+\sin x\left)\right)+\ln c$

$\Rightarrow y=\frac{c}{2+\sin x}-1$

Since $y\left(0\right)=1\Rightarrow c=4$

Hence $y\left(\frac{\pi }{2}\right)=\frac{4}{3}-1=\frac{1}{3}$