Chapter 3 - Systems of ordinary differential equations

Abstract

We extend the existence and uniqueness theorems of Chapter 2 to systems. Then we develop the notion of the fundamental matrix as a solution and utilize it to write solutions of non-homogeneous systems and obtain variation of parameters for systems. Assuming that the matrix coefficient is constant, we discuss the concept of exponential matrix and its development and applications. We briefly discuss Jordan canonical forms and their usage. We state and prove the Putzer algorithm and use it to calculate the exponential matrix, regardless of the nature of the eigenvalues of the constant coefficient matrix. Toward the end of the chapter, we discuss perturbed linear and nonlinear systems and represent corresponding solutions in the form of variation of parameters. We end the chapter with a general discussion on how to compute the fundamental matrix.