Let $y' = g(t,y)sin^2(y)$, with $ g: \mathbb{R} \times \mathbb{R}\to \mathbb{R}$, g differentiable.
I want to prove that if a solution is maximal, there exists the limits $\lim_{t\to \pm \infty} y(t)$.
This is what I have done: product of differentiable functions is still differentiable, so I can say that there exists a local solution and it is unique.Then there also exists a maximal solution that is unique.
To try and calculate that limit I must also say that the maximal solutions are globally defined on all $\mathbb{R}$, which is where I am stuck. Besides, even assuming that I know this fact, I would not know how to continue.
Any hint?