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: , 3
Solution:
$\frac{d y}{d x}+2 y \tan x=2 \sin x$
I.F. $=e^{\int 2 \tan x d x}=\sec ^{2} x$
The solution of the differential equation is
$y \times$ I.F. $=\int$ I.F $\times 2 \sin x d x+C$
$\Rightarrow y \cdot \sec ^{2} x=\int 2 \sin x \cdot \sec ^{2} x d x+C$
$\Rightarrow y \sec ^{2} x=2 \sec x+C$......(1)
When $x=\frac{\pi}{3}, y=0 ;$ then $C=-4$
$\therefore$ From (1), $y \sec ^{2} x=2 \sec x-4$
$\Rightarrow y=\frac{2 \sec x-4}{\sec ^{2} x} \Rightarrow y\left(\frac{\pi}{4}\right)=\sqrt{2}-2$