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One approach to convert this into an ODE is with index reduction methods. These allow you to convert high-index DAEs into low-index DAEs or ODEs. See section VII.2 of "Solving Ordinary Differential Equations II" by Hairer and Wanner. Consider a generic, Hessenberg index-2 DAE \begin{align} y' &= f(y, z) \\ 0 &= g(y) \end{align} ...