A simple ordinary differential equation - MathOverflow

A simple ordinary differential equation

2010-11-15T08:21:20Z

Consider an entire function $f : \mathbb{C} \rightarrow \mathbb{C}$! We search the function $$ g: (a,b) \rightarrow \mathbb{C},$$ which solves the following equation locally: $g'(t)=f(g(t))$ and $g(0)=f(x_0)$.

I can compute the inverse $G$ of $g$, if $f(x_0) \neq 0$, i.e. $$ G(y) = \int\limits_{f(x_0)}^y \frac{d s}{f(s)}.$$

I also known how to compute the Taylor expansion recursively, whose radius of convergence is positive (see below). Also we can give suitable approximations of the solution in terms of Picard iterations. I am not interested in such a solution!

Is there an alternative to this integral expression?

Answer by Florian for A simple ordinary differential equation

2010-11-15T10:41:39Z

The power series for $g$ has a positive radius of convergence; this is a consequence of the Cauchy-Kovalevskaya theorem (which is a statement about PDEs, but an ODE is just a PDE with one variable).

Answer by Michael Renardy for A simple ordinary differential equation

2010-11-15T13:00:32Z

You don't need the Cauchy-Kovalevskaya theorem. Just the analytic inverse function theorem.

Answer by Dick Palais for A simple ordinary differential equation

2010-11-15T15:41:41Z

It is hard to guess what you are looking for. Take the apparently simpler case where $f$ is a polynomial, say of degree $d$. If $d = 1$ you have an explicit solution in terms of the exponential function (because your $G$ is logarithmic). If $d = 2$ the solution can be written in terms of trigonometric functions. If $d = 3$ you need elliptic functions to express the solution explicitly. As soon as $d$ is greater than $3$, I don't know of any standard naming for the functions you get or any interesting theory of these functions.