I need a hint for the following problem. It seems that separating variables does not work. Please leave just a hint not a whole solution.
Solve the Cauchy problem. $y'(x)=\frac{x^{2}}{2+\sin(x^{2})}\cos((y(x))^{2})\;\; and\;\; y(0)=0 .$
(a) Prove that there exist one and only one solution defined for every $x\in \mathbb{R}$.
(b) Prove that $y(x)\to \sqrt{\pi/2}$ as $x\to +\infty.$