Reference for analytic dependency on inititial values for ODE with analytic coefficients

Question

I would like to see a book where it is proven that any solution to a given $2$nd order ODE with analytic coefficients depends analytically on the initial values. For example in "differential equations with applications and historical notes" by George Simmons it is only proven, that any solution is analytic. They don't talk about dependence on the initial values.
Thanks in advance!