On Sun, 12 May 2013 16:02:14 +0100, José Carlos Santos <firstname.lastname@example.org> wrote:
>Hi all, > >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.