Let $x $ be a solution of $(E) $. Show that $x $ is Lipschitz continuous?

Question

We consider differential equation $$ (E):\qquad x'(t)=\cos(2\pi(x(t)-t)) $$

Let $x $ be a solution of $(E) $. Show that $x $ is Lipschitz continuous.

An idea please.

Answer 1

A ($C^1$-derivable) function is $k$-Lipschitz continuous iff the absolute value of the derivative is majored by $k$ (ie, its normal derivative is bounded by $-k$ and $k$). Here, $x'(t)$ is a $cos$ of something, so its values are necessarily between $-1$ and $1$. This means $x(t)$ is $k$-Lipschitz continuous for $k = 1$ and above. There might be a better bound $k$ to find, though.