I have a problem with assesing the accuracy of my numerical calculation. I have a 2nd order ODE. It is an eigenvalue problem of the form:$$ y'' + ay' + \lambda^2y = 0, $$and the boundary condiations are:$$ y(0) = y(1) = 0. $$This equation describes a vibrating string, clamped at x=0 and x=1, with a certain mass distribution. I want to be able to calculate the eigenvalues of this equation numerically. I do this by using the Runge-Kutta method to find solutions to the equation with initial conditions:$$ y(0) = 0, y'(0) = 1 $$with different values of $\lambda$ and then looking for the ones that are zero at x=1.I terminate my search for eigenvalues when I find a function that has$$y(1) < \delta,$$where $\delta$ is a predefined constant. Now the problem is that I'd like to know how accurate my calculations are, i.e. how do I choose the values of $\delta$ and the stepsize used in computing the values of the potential eigenfunction, so that the error in the eigenvalue is less than, say, $ 10^{-3}$? 
Accuracy of numerical methods in finding eigenvalues
ode;eigenvalues;runge kutta;numerical;accuracy