We study initial value problems having dynamics ruled by discontinuous ordinary differential equations with the property of possessing a unique solution. We identify a precise class of such systems that we call \emph{solvable intitial value problems} and we prove that for this class of problems the unique solution can always be obtained analytically via transfinite recursion. We present several examples including a nontrivial one whose solution yields, at an integer time, a real encoding of the halting set for Turing machines; therefore showcasing that the behavior of solvable systems is related to ordinal Turing computations.

翻译：我们研究具有不连续常微分方程规则且具有唯一解性质的初值问题。我们确定了这类系统的一个精确类别，称之为“可解初值问题”，并证明对于这类问题，其唯一解总能通过超限递归分析求解。我们给出了若干例子，包括一个非平凡的例子，其在整数时刻的解给出了图灵机停机集的实数编码；从而展示了可解系统的行为与序数图灵计算相关。