Differential Equations: An Introduction to Modern Methods and Applications, 2nd Edition

Systems of First Order Linear Equations

n this chapter, we build on elementary theory and solution techniques for first order linear systems introduced in a two-dimensional setting in Chapters 3 and 4. Science and engineering applications that possess even a modest degree of complexity often lead to systems of differential equations of dimension n > 2. The language, concepts, and tools of linear algebra, combined with certain elements of calculus, are essential for the study of these systems. The required mathematical background from matrices and linear algebra is presented in Appendix A. A reader with little or no previous exposure to this material will find it necessary to study some or all of Appendix A before proceeding on to Chapter 6. Alternatively, the reader may take up the study of Chapter 6 straightaway, drawing on necessary results from Appendix A as needed.

A large portion of Chapter 6 generalizes the eigenvalue method presented in Chapter 3 to constant coefficient linear systems of dimension n > 2. Since carrying out eigenvalue calculations by hand for matrices of dimension greater than 2 is time-consuming, tedious, and susceptible to error, we strongly recommend using a computer or a calculator to perform the required calculations, once an understanding of the general ...