A function y=f(x) satisfies f(x)2x+ x-(1+^2x)f'(x)=0, f(0)=0.

Question

A function $y=f(x)$ satisfies $f(x)\sin2x+\sin x-(1+\cos^2x)f'(x)=0$, $f(0)=0$. Then $f\!\left(\dfrac{\pi}{2}\right)$ is:

Accepted Answer

Rewriting: $(1+\cos^2x)f'(x)-2\sin x\cos x\,f(x)=\sin x$, a linear ODE in $f$. With I.F. related to $(1+\cos^2x)$, solving and applying $f(0)=0$ yields $f(\pi/2)=1$.

Solutions · 1