Differential algebraic equations with properly stated leading terms

In this paper, we study differential algebraic equations (DAEs) of the form A(χ, t)(d(χ, t))′+ b(χ, t) = 0 with in some sense well-matched matrix functions A(χ, t) and D(χ, t) := d′χ (χ, t) as they arise, e.g., in circuit simulation. We characterize index 1 DAEs in this context. After analyzing those index 1 equations themselves, we apply Runge-Kutta methods and BDFs, provide stability inequalities, and show convergence. The cases of the image space of D(χ, t) or the nullspace of A(χ, t) remaining constant are pointed out to be essentially favourable for the qualitative behaviour of the approximations on long intervals. Hence, when modelling with DAEs one should try for those, constant subspaces. Relations to quasilinear DAEs in standard formulation E(χ, t)χ′ + ƒ (χ, t) = 0 are considered, too.