Effective Computability of Solutions of Ordinary Differential Equations The Thousand Monkeys Approach

In this note we consider the computability of the solution of the initial-value problem for ordinary differential equations with continuous right-hand side. We present algorithms for the computation of the solution using the “thousand monkeys” approach, in which we generate all possible solution tubes, and then check which are valid. In this way, we show that the solution of a differential equation defined by a locally Lipschitz function is computable even if the function is not effectively locally Lipschitz. We also recover a result of Ruohonen, in which it is shown that if the solution is unique, then it is computable, even if the right-hand side is not locally Lipschitz. We also prove that the maximal interval of existence for the solution must be effectively enumerable open, and give an example of a computable locally Lipschitz function which is not effectively locally Lipschitz.