# Accuracy of Runge-Kutta method when no analytical solution

1. Jan 5, 2012

### jimmychoo

1. The problem statement, all variables and given/known data
I have to find eigenvalues to
$$\frac{d^2y}{dx^2} + p^2 e^x y = 0,\, y(0)=0,y(1)=0$$
using the Runge-Kutta single step method to solve the ODE (splitting it up), with step length $h$and then another numerical method. This is not a problem. However, I need to be able to find the eigenvalues to a specified accuracy: so this is my question:

Is it possible to bound the error of the Runge-Kutta method (at a certain step length) for solving this ode without using the analytical solution?

3. The attempt at a solution
Runge-Kutta is a 4th order method, so I tried to model the error as $kh^5$ and using different step lengths solve for the constant, but this didn't work.
Maybe I'm just missing an analytical feature of this particular equation (if that's the case and anybody else can see it, don't give me the specifics of it!)

thanks for any help!