I am faced with a non-autonomous initial value problem for a function $x:[0,\infty) \to \mathbb{R}^2$ of the form $$ x'(t) = f(t,x(t)) $$ for $f: [0,\infty) \times \mathbb{R}^2 \to \mathbb{R}^2$ with initial condition $x(0)$. Now, the function $f$ is such that $$ f(t,x) = t^{-1} f_{-1}(x) + f_0(x) + t f_1(x) $$ where the functions $f_n: \mathbb{R}^2 \to \mathbb{R}^2$ are analytic (in fact, polynomial). Furthermore, the initial condition is such that $$\lim_{t\to 0^+} f(t,x(t))$$ exists. This means that there is a formal power series for $x(t)$ around $t=0$ which solves the initial value problem; although it depends on a parameter (related to $x'(0)$) which cannot be fixed. In other words, I get a formal power series in $ct$, for some real number $c$ which cannot be determined. My problem is to determine the radius of convergence of this power series in $ct$.

Alas, my expertise with analytic solutions of ODEs stops with the standard undergraduate fare of the Frobenius method,... but only for linear equations. Hence I am asking the MO community for some readable reference(s) for the nonlinear case.

Thanks in advance.

**Added** (in response to comments below and particularly KConrad's answer)

What I am actually interested is in whether the solution will blow up in finite time. (Actually, in the problem $t$ is not really time, but inverse distance from a black-hole-like singularity and by blowing up in finite time, what I am after is whether the solution is indeed a black hole; i.e., whether there is an event horizon.) I cannot prove that this is the same as the formal power series solution having a finite radius of convergence, but this is precisely what happens in the black hole solutions I know: Schwarzschild and Reissner-Nordström, for instance.

notnecessarily equal the maximum existence time? $\endgroup$ – KConrad May 28 '10 at 22:14