Search any question & find its solution
Question:
Answered & Verified by Expert
If $y=y(x)$ and $\frac{2+\sin x}{y+1}\left(\frac{d y}{d x}\right)=-\cos x, y(0)=1$, then $y\left(\frac{\pi}{2}\right)$ is equal to
Options:
Solution:
2101 Upvotes
Verified Answer
The correct answer is:
$\frac{1}{3}$
$\begin{aligned} & \frac{2+\sin x}{y+1} \frac{d y}{d x}=-\cos x \\ & \Rightarrow \int \frac{d y}{y+1}=\int \frac{-\cos x d x}{2+\sin x} \\ & \Rightarrow \log _{\mathrm{e}}|y+1|=-\log _{\mathrm{e}}|2+\sin x|+\log _{\mathrm{e}} C \\ & \Rightarrow \log _e|y+1|+\log _e|2+\sin x|=\log _e C \\ & \Rightarrow \log _e|(y+1)(2+\sin x)|=\log _e C \\ & \Rightarrow(y+1)(2+\sin x)=C \\ & \text { Putting } x=0 \text { and } y=1 \text { we get } c=4 \\ & \Rightarrow(y+1)(2+\sin x)=4 \\ & \end{aligned}$
Now putting $x=\frac{\pi}{2}$ we get $y=\frac{1}{3}$
Now putting $x=\frac{\pi}{2}$ we get $y=\frac{1}{3}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.