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Effective computability of solutions of ordinary differential equations: the thousand monkeys approach
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In this note we consider the computability of the solution of the initial-
value problem for ordinary di erential 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 possi-
ble solution tubes, and then check which are valid. In this way, we show
that the solution of a di erential equation de ned by a locally Lipschitz
function is computable even if the function is not e ectively locally Lips-
chitz. 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 e ectively enumerable open, and give an example
of a computable locally Lipschitz function which is not e ectively locally
Lipschitz.