>> 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? > > I suspect you may never get the sort of answer you want here. > There are those numerical methods out there. The proof that > they work typically depends on certain hypotheses, which > hypotheses typical entail the existence and uniqueness of > the solution. You have a DE that does not satisfy those > hypotheses, so you shouldn't expect those methods to > give the solution you want.