Consider the ODE:$$ y'= sin(y)$$I have to investigate the stability and asymptotic stability of the equilibrium point $\tilde{y}$=0. Now we have defined an equilibrium point stable, in the sense of Ljapunov, if :$$\forall\epsilon\ \exists\delta>0\ : \| y(t_0)-\tilde{y}\|<\delta \rightarrow \|y(t)-\tilde{y}\|<\epsilon\ ,\forall t>t_0$$ and asymptotically stable, if there exists a neighbourhood $U_\tilde{y}$ of $\tilde{y}$ s.th. from $y(t_0)\in U_\tilde{y} $ it follows $\lim_{t\rightarrow\infty}y(t)=\tilde{y}$.
The problem I have right now is that solving the equation brings me to $$ \csc y - \cot y=e^{x+c_1}$$ and that I do not know how to go on from there.
It was said to be supposedly easy to check stability and asmyptotic stability but I do not see how I can find such $\delta$ for every $\epsilon$ 'easily'. Any help on how to check whether $\tilde{y}=0$ is stable or not would be greatly appreciated!