A function y = f(x) satisfies f(x)sin2x + sinx − (1+cos^2x) f′(x) = 0 with condition f(0) = 0.

Question

A function \(y=f(x)\) satisfies \(f(x) \sin 2 x+\sin x-\left(1+\cos ^{2} x\right) f^{\prime}(x)=0\) with condition \(f(0)=0\). Then \(f\left(\frac{\pi}{2}\right)\) is equal to

(1) 1

(2) 0

(3) -1

(4) 2

Answer 1

Correct option is (1) 1

\(\mathrm{\frac{d y}{d x}-\left(\frac{\sin 2 x}{1+\cos ^{2} x}\right) y=\sin x}\)

I.F. \(=1+\cos ^{2} \mathrm{x}\)

\(\mathrm{y \cdot\left(1+\cos ^{2} x\right)=\int(\sin x) d x}\)

\(=-\cos \mathrm{x}+\mathrm{C}\)

\(\mathrm{x}=0, \mathrm{C}=1\)

\(\mathrm{y}\left(\frac{\pi}{2}\right)=1\)