Let y=y(x) be the solution of the differential

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 :

  1. $\sqrt{2}-2$

  2. $\frac{1}{\sqrt{2}}-1$

  3. $2-\sqrt{2}$

  4. $2+\sqrt{2}$


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 $x=\frac{\pi}{4}$

$y \cdot 2=2 \sqrt{2}-4$

$y=\sqrt{2}-2$

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