12 - Matrices and Systems of Differential Equations

Abstract

This chapter describes matrices as a powerful tool to discuss the behavior of solutions to systems of differential equations. If A(t) and f(t) are continuous matrix and vector functions, respectively, on some interval [a, b], that is, the component functions of both A(t) and f(t) are continuous, then there exists a unique vector function φ(t) that is a solution to the initial value problem on the entire interval [a, b]. Fundamental matrix solutions are not unique, because a solution vector may be multiplied by any constant and still remain a solution. The calculation of a fundamental or principal matrix solution is generally impossible if A(t) is nonconstant.