Question

If $y=y(x), y \in\left[0, \frac{\pi}{2}\right)$ is the solution of the differential equation $\sec y \frac{d y}{d x}-\sin (x +y)-\sin (x-y)=0$, with $y(0)=0$ then $5 y^{\prime}\left(\frac{\pi}{2}\right)$ is equal to______.

Answer 1

$\sec y \frac{d y}{d x}=2 \sin x \cos y$
$\sec ^{2} y d y=2 \sin x d x$
$\tan y=-2 \cos x+c$
$c=2$
$\tan y=-2 \cos x+2$
$\Rightarrow$ at $x=\frac{\pi}{2}$
$\tan y=2$
$\sec ^{2} y \frac{d y}{d x}=2 \sin x$
$5 \frac{d y}{d x}=2$