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,