global and unique solution for ODE 
Let $y' = f(x) p(\cos y) + g(x) q( \sin y) $ and $y(0) =1$, with $f,g$ continuous and $p,q$ polynomials. Then there exists unique solution on all $\mathbb{R}$ ?

Its easily solved if i assumed that $f,g$ are bounded, but the question asks about $f,g$ that are not necessarily bounded.
Can any one give a hint or a solution !

$f$ and $g$ are bounded on every interval $[-R,R]$ for any $R=N\in\Bbb N$. Thus you get unique solutions over these intervals, and by the uniqueness the solution for $R=N+1$ extends the solution for $R=N$. In the limit this gives a unique solution over all of $\Bbb R$.
