Consider the ODE
$$ u'(t) = f(t,u,p), \qquad u(0) = v $$
where $p$ is a control parameter, and let $u(t;v,p)$ denote the solution to the problem above for fixed $v$ and $p$.
It is apparently "well known", that the analyticity of $f$ implies that $x$ is analytic with respect to the parameter $p$. More precisely, if $f(t,u,p)$ is real-analytic in the open set $(t_0-a,t_0+a) \times (u_0-b,u_0+b) \times (p_0-c,p_0+c)$, then the mapping $$\varphi: (t,v,p) \to u(t;v,p)$$ is real-analytic in an open subset of $(t_0-a,t_0+a) \times (u_0-b,u_0+b) \times (p_0-c,p_0+c)$.
If I understand this correctly, I should be able to find an open set $B$ containing $(t_0,u_0,p_0)$ and constants $D,S > 0$ such that
\begin{equation} | \partial^k_p \phi(t,v,p)| \leq D \frac{k!}{S^k} \qquad (t,v,p) \in B \qquad k \in \mathbb{N} \end{equation}
The problem is that, in spite of this result supposedly being known since Cauchy first proved it, I am struggling to find a proof of how $B$, $D$ and $S$ are constructed. In particular, the analyticity of $f$ implies the existence of constants $C, R$ such that $$ | \partial^{\alpha_t}_t \partial^{\alpha_u}_u\partial^{\alpha_p}_p f (t,u,p) | \leq C \frac{\alpha_t!\alpha_u! \alpha_p!}{R^{\alpha_t+\alpha_u+\alpha_p}}, \qquad (t,u,p) \in (t_0-a,t_0+a) \times (u_0-b,u_0+b) \times (p_0-c,p_0+c), \qquad \alpha_t, \alpha_u, \alpha_p \in \mathbb{N} $$ and I would like to find $B$, $D$, $S$ in terms of $C,R$, $a,b,c$.
I have seen several references where the analyticity of $\varphi$ is proved in the "time" variable $t$, and hence estimates for $\partial_t^k \varphi$, are found, but I fail to adapt them to the case of analyticity with respect to parameters or initial condition. Also, I often read the suggestion of "adding the equation $p'=0$ to the problem", but this route then implies one is able to prove analyticity with respect to initial conditions, and hence able to find a bound on $\partial_v^k \varphi$, which is still different from $\partial_t^k \varphi$.
My attempts to ask clarifications to authors that deal with the "time" analyticity have been ignored.
In an attempt to write down the estimates, I have derived an evolution equation for $\partial_p^k \varphi$, but I do not manage to obtain the desired estimate
Can anyone help, or provide a reference, or a set of notes where these bounds are found explicitly?
