I am interested, how a modelling tool (in my case OpenModelica) solves the following systems. In advance, I know a little bit the theory behind transforming a differential algebraic equation system into an ODE (keyword "index-reduction").

How do these tools solve a system of equations without differential equations? Is the system nevertheless transformed (index reduction) into an ODE?

E.g. if I use the DASSL Solver: Let's assume there are a few ODEs, some algebraic equations (where some variables are also in the ODEs) and there are also some algebraic equations that are not coupled with the ODE. So as I understand it, this would lead to a DAE where the solver would perform index reduction on the algebraic equations that are coupled with the ODE, and Newton’s method solves the system of equations that are not coupled also in every time-step