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Just for completeness, don't forget about one of @Artes' favourite functions: FrobeniusSolve. The Pick statements select solutions with positive integers less than or equal to p. kTuples[m_Integer, k_Integer, p_Integer] := Block[{s = FrobeniusSolve[ConstantArray[1, k], m]}, s = Pick[s, UnitStep[Apply[Sequence, Table[p - s[[All, i]], {i, k}]]], ...


One can use Picard-type iteration to get the solution: Using an approximation to x'[t] (in the integral), we can integrate the ODE to obtain a new approximation. Remarkably, it converges in just two steps. My original thought was to step through the integration using the tools from tutorial/NDSolveStateData to build an interpolation of x'[t] at each step ...

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