This question is perhaps too vague to have a meaningful answer, but here it goes.
In what follows, I am only interested in functions defined in some interval of the type [0,a], with a > 0.
Suppose that I want to solve numerically the ODE f'(x) = 2*sqrt(f(x)), under the condition f(0) = 0. Of course, the null function is a solution of this ODE. The problem is that I am not interested in that solution; the solution that I am after is f(x) = x^2.
For my purposes, numerical solutions are enough, but if I try to solve numerically an ODE of the type f'(x) = g(f(x)) (with g(0) = 0) and f(0) = 0, what I get is the null function. So far, my way of dealing with this has been to solve numerically the ODE f'(x) = g(f(x)) and f(0) = k, where _k_ is positive but very small and to hope that the solution that I get is very close to the solution of the ODE that I am interested in (that is, the one with k = 0). Do you know a better way of dealing with this problem?