On nonlinear algebraic differential systems reducible to non-degenerate systems of ordinary differential equations. Theory and numerical methods of solution

In this paper, we consider algebraic differential systems of the form
$$ \frac{dAx}{dt}=Bx+f(x,t) $$
with a regular pair of matrices $(A,В)$. The conditions of reducibility of such systems to non-degenerate systems of ordinary differential equations (ODE) of first order with respect to the derivative $x'(t)$ are given. Methods for the numerical solution of $x(t)$ are proposed.