Question

If $y=y\left(x\right)$ and $\frac{2+\sin x}{y+1}\left(\frac{dy}{dx}\right)=-\cos x,y\left(0\right)=1$, then find the value of $y\left(\frac{\pi }{2}\right)$.

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

Answer 1

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

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

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

Integrating both sides $\Rightarrow \ln (1+y)=-\ln (2+\sin x)+c$

Substitute $x=0$ and $y=1$

$\Rightarrow \ln 2=-\ln 2+c\Rightarrow c=\ln 4$

Substitute $x=\frac{\pi }{2},\ln (1+y)=\ln 3+\ln 4=\ln \frac{4}{3}\Rightarrow y=\frac{1}{3}$