A function y=f(x) satisfies [f(x) 2 x+ x-(1+ ^2 x) 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

Accepted Answer

$$\begin{aligned}
& \frac{d y}{d x}-\left(\frac{\sin 2 x}{1+\cos ^2 x}\right) y=\sin x \
& \text { I.F. }=1+\cos ^2 x \
& y \cdot\left(1+\cos ^2 x\right)=\int(\sin x) d x \
& =-\cos x+C \
& x=0, C=1 \
& y\left(\frac{\pi}{2}\right)=1
\end{aligned}$$

Solutions · 1