I have to prove that for the equation:
$$y'(x)=1+\sin^2(y(x))$$
I started notice that $y'(x)$ is a limited, continuos and $C^{\infty}$ function defined on $R^{2}$. Due to it's limitation I can intuitively say that it's a Lipschitz continuous function, so there exist both global and local solutions.
Could someone help me to say this stuff in a more advanced math language, proving the Lipschitz conditions? Thank you!! :)
$f(t,y)=1+\sin^2 y$ is differentiable and thus locally Lipschitz, which provides uniqueness.
As $f$ and thus $y'$ is bounded, you get that $y$ is bounded over any finite interval, which proves that any solution can be extended without bound, i.e., any maximal solution has all of $\Bbb R$ as domain.