Further results on solvability of a singular system of ordinary differential equations (Q2366415)

This paper examines the differential equation \(Ax'=F(x)\) under the initial assumption that \(A\) has constant rank, nullspace, and range. A constant coordinate change converts the differential equation to the two equations \(0=F_ 1(x_ 1,x_ 2)\) and \(x_ 2'=\tilde A^{-1}F_ 2(x_ 1,x_ 2)\). If \(G=\partial F_ 1/\partial x_ 1\) is nonsingular, the system is index one and easily solved. If \(G\) is singular, the author assumes that \(F_ 1(x_ 1,x_ 2)=0\) is the linear constant coefficient equation \(Gx=0\), and repeats the process on the system made up of \(Gx'=0\) and \(x_ 2'=\tilde A^{-1}F_ 2(x_ 1,x_ 2)\). Implicit systems of differential equations, or DAEs as they are frequently called, have been widely studied in the literature. The assumptions in this paper severely restrict the types of nonlinearities considered. Considerably more general results, which include those in this paper as a special case, may be found in the recent papers by P. Rabier, S. Reich, and W. Rheinboldt.