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Let y = y(x) be the solution of the differential equation \cos x\frac{dy}{dx} + 2y\sin x = \sin 2x, \ x\in \left(0,\frac{\pi}{2} \right ) . If y\left(\frac{\pi}{3} \right ) = 0 then y\left(\frac{\pi}{4} \right ) is equal to:
Option: 1 2 +\sqrt2
Option: 2 \frac{1}{\sqrt2} - 1
Option: 3 2 - \sqrt2
Option: 4 \sqrt2 - 2

Answers (1)

best_answer

\\\cos x \frac{d y}{d x}+2 y \sin x=\sin 2 x \\\\ \Rightarrow \frac{d y}{d x}+2 y \tan x=2 \sin x

\\ I.F.=e^{2 \int \frac{\sin x}{\cos x} d x} \\\\ {\qquad=e^{\ln \sec ^{2} x}=\sec ^{2} x }

\\\Rightarrow y \times \sec ^{2} x=2 \int \sin x \times \sec ^{2} x d x \\ \Rightarrow y \sec ^{2} x=\frac{2}{\cos x}+c \\ 0=4+c \Rightarrow c=-4

\operatorname{at} x=\pi / 4 \\ \\y \times 2=2 \sqrt{2}-4 \\ \\\Rightarrow y=\sqrt{2}-2

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