Solution of Linear System of Differential Equations By Rational Canonical Form

Abstract

Solving linear system of differential equations by Jordan canonical form needs the change in the real field to complex and then retrieve the complex solutions to the real ones. B. Malesevic, D. Todoric, I. Jovovic, and S. Telebakovic suggest that it is more convenient to apply the rational canonical form than the Jordan canonical form. They reduce the linear system $$DY=AY$$ to higher order differential equations $$f_j(D)z=0$$ , where the polynomials $$f_1,f_2,\ldots ,f_k$$ appear in the rational canonical form of A. ( $$D=d/dx$$ .) The present paper proves the converse of their theorems and discovers certain dimensionality relations between the solution spaces of the two types of differential equation problems. The approach is pure algebraic and leaves the door open for extensions to formal differential equations.