Exercise 2.3.4: In each part below make up an autonomous ODE $u′ = f(u)$ (by finding a suitable function $f$) that has the indicated fixed points with the indicated stability.
Fixed points $u = k\pi$ for all integers $k$, unstable when $k$ is even, stable when $k$ is odd.
The required function $f$ is the sine: $$ u'=\sin u. $$ Indeed, $f(u)=0$ iff $u=k\pi$, $k\in \mathbb Z$, so $u=k\pi$ are the fixed points; $f'(u)=\cos x$, thus $f'(k\pi)>0$ when $k$ is even and $f'(k\pi)<0$ when $k$ is odd.