CHAPTER 6

SYSTEMS OF DIFFERENTIAL EQUATIONS

In this chapter we motivate, by means of a simple observation, the generalization to systems of equations of what we have learned for scalar differential equations.

Consider the initial value problem for a system of differential equations

(6.1) equation

where y is an N-component vector and f is an N-component vector-valued function:

equation

The simplest systems are linear systems with constant coefficients:

(6.2) equation

Here A is a constant N x N-matrix and F has N components.

Let us assume that A has a complete set of eigenvectors; that is, there exists a transformation S that transforms A into a diagonal matrix λ:

equation

Exercise 6.1 The exponential of a matrix A NxN is defined via the Taylor series,

equation

(a) Show that exp(A) is well defined (i.e., the series is always convergent). ...

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