Question

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

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

Answer 1

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

$\frac{dy}{dx}=\frac{-(y+1)\cos x}{2+\sin x}$
$\Rightarrow \int \frac{dy}{y+1}=-\int \frac{\cos x}{2+\sin x}dx$
$=\ln (y+1)=-\ln (2+\sin x)+c$
$=(y+1)(2+\sin x)=C;$
for $x=0,y=1\Rightarrow C=4$
$(y+1)(2+\sin x)=4$
$y\left(\frac{\pi }{2}\right)=\frac{1}{3}$