JEE Mains · Maths · STD 12 - 9. differential equations
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 \(y (\frac{\pi}{3})=0,\) then \(y (\frac{\pi}{4})\) is equal to
- A \(\sqrt{2}-2\)
- B \(\frac{1}{\sqrt{2}}-1\)
- C \(2-\sqrt{2}\)
- D \(2+\sqrt{2}\)
Answer & Solution
Correct Answer
(A) \(\sqrt{2}-2\)
Step-by-step Solution
Detailed explanation
\(\cos x \frac{d y}{d x}+2 y \sin x=\sin 2 x\) \(\frac{d y}{d x}+\frac{2 \sin x}{\cos x} y=2 \sin x\) I.F. \(=e^{\int \frac{2 \sin x}{\cos x} d x}\) \(=e^{2 \ln \sec x}=\sec ^{2} x\) \(y \cdot \sec ^{2} x=\int 2 \sin x \cdot \sec ^{2} x d x\)…
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