# 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

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

The simplest systems are linear systems with constant coefficients:

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:

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

*(a) Show that*exp(

*A*)

*is well defined (i.e., the series is always convergent). ...*

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