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

Question

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

Accepted Answer

$\begin{gathered} \frac{dy}{dx}-(\frac{\sin 2x}{1+{\cos }^{2}x})y=\sin x \ =1+{\cos }^{2}x \ y\cdot (1+{\cos }^{2}x)=\int (\sin x)dx \ =-\cos x+C \ x=0,C=1 \ y(\frac{\pi }{2})=1 \end{gathered}$

Solutions · 1