Theorem 1: For any constant vector c,


is a solution of system (2). Moreover, the solution of (2) given by

satisfies the initial condition x(0)=x₀.

Proof: Evidently



and

Definition 1: The matrix e^At is called the principal matrix solution of the system (2).
General solution algorithm using linear algebra software:

1. compute e^At

2. compute the general solution of system (2) by (64) or solution to the initial value problem by (65).

Example 1: Consider the system

find the general solution.

1. Computing e^At

2. computing the general solution of system (1) by (64)

Exercises

1. Solve the initial value problem