Show that a differential equation satisfies Lipschitz condition

Prove that if $$\frac{dx}{dt}=(3t^2+1)\cos^2(x)+(t^2-2t)\sin (2x)=f(t,x),$$ then $f(t,x)$ satisfied Lipschitz condition on the strip $S_{\alpha}=\{(t,x):|t|\le\alpha , |x|\le \infty , \alpha >0\}$.

Can I get some help for the above problem please.

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To show that a function is Lipschitz on $S_\alpha$, it suffices to check that its partial derivatives are bounded on $S_\alpha$. See Bounded partial derivatives imply continuity

Here $f_x = -(3t^2+1)\sin 2x+2(t^2-2t)\cos 2x$ and $f_t=6t\cos^2x+(2t-2)\sin 2x$.
Estimate the absolute value of each derivative using (a) triangle inequality; (b) the fact that sine and cosine are bounded by $1$; (c) the given bound $|t|\le \alpha$.

By the way, if you just want to apply the uniqueness/existence theorem, then only the Lipschitz condition in $x$ is needed. For the $t$ variable, continuity suffices.