Approximated solutions in rational form for systems of differential equations (Q1817788)

The authors study systems of linear first order differential equations \[ Y'(t)=A(t)Y(t)+b(t), \] with \(A:D\rightarrow {\mathbb C}^{n\times n}\), \(b,Y:D\rightarrow {\mathbb C}^{n\times 1}\) where \(D\subset {\mathbb R}\). They restrict their attention to the case that the elements of \(A\) and \(b\) are analytic functions (as \(D\subset {\mathbb R}\), probably real-analytic). After recalling known results on systems of ODE's, they set out to study the question: when does the system have rational solutions? As the fundamental matrix of a homogeneous system has a basis for the solution space as columns, answering the question turns out to have to identify matrix rational functions. To this end they construct a table of ranks of connected matrices (reminiscent of the same type of tables that arise in matrix Padé approximation). The result is (not surprisingly) that rationality is connected with the appearance of an infinite block in the rank table. Several examples are discussed, including the reduction of the partial differential equation situation to a system. No proofs are given.