Under which conditions a solution of an ODE is analytic function?

If I'm not wrong there is a theorem that says that if the conditions for Picard's theorem are satisfied, for an ode $\dot x=f(x,t)$, then the solution of the ode is as smooth as $f$. I think I'm not wrong with this fact.

So if $f$ is $\mathcal{C}^k$ then $x(t)$ will be also $\mathcal{C}^k$. I wonder if the fact that $f$ is analytic implies also that $x$ is analytic or if there is another condition that implies so.

Solution 1:

Yes, this is called Cauchy's theorem. If $f(x,t)$ is analytic at $(x_0,t_0)$ then there exists a unique solution analytic at the point $t_0$. The same theorem is true when you consider both $x,t\in\mathbf C$.

Your statement about the smoothness of solutions is not quite correct. You have that if $f\in C^k$ then solution is $C^{k+1}$.