Question:
Let y=y(x) be the solution of the differential
equation $\cos x \frac{d y}{d x}+2 y \sin x=\sin 2 x$,
$x \in\left(0, \frac{\pi}{2}\right) .$ If $y(\pi / 3)=0$, then $y(\pi / 4)$ is equal
to :
Correct Option: 1
Solution:
$\cos x \frac{d y}{d x}+2 y \sin x=\sin 2 x$
$\frac{d y}{d x}+\frac{2 \sin x}{\cos x} y=2 \sin x$
I.F. $=e^{\int 2 \frac{\sin x}{\cos x} d x}$
$=e^{2} \ln \sec x=\sec ^{2} x$
$y \cdot \sec ^{2} x=\int 2 \sin x \cdot \sec ^{2} x d x$
$y \sec ^{2} x=2 \int \tan x \sec x d x$
$y \sec ^{2} x=2 \sec x+c$
At $x=\frac{\pi}{3}, y=0$
$\Rightarrow 0=2 \sec \frac{\pi}{3}+C \Rightarrow C=-4$
$y \sec ^{2} x=2 \sec x-4$
Put $x=\frac{\pi}{4}$
$y \cdot 2=2 \sqrt{2}-4$
$y=\sqrt{2}-2$