Sampling the initial conditions for differential-algebraic equations

To apply the numerical methods developed for systems of ordinary differential equations with a degenerate matrix in front of the principal part, it is necessary to set the initial condition. In contrast to the systems of ordinary differential equations in normal form, this condition cannot be set arbitrarily, and it should be consistent with the right part of the system. An algorithm is proposed for sampling of deficient initial conditions for the case when the initial conditions are given not in the standard way, but represent the Showalter-Sidorov condition. This algorithm is based on some facts from the theory of projectors. In the article we have illustrated the results by a simple example.