We show how linear systems can be written in matrix form, and we make many comparisons to topics we have studied earlier.

### Linear Systems of Differential Equations

A first order system of differential equations that can be written in the form

is called a*linear system*.

The linear system (eq:10.2.1) can be written in matrix form as

or more briefly as

where We call the*coefficient matrix*of (eq:10.2.2) and the

*forcing function*. We’ll say that and are

*continuous*if their entries are continuous. If , then (eq:10.2.2) is

*homogeneous*; otherwise, (eq:10.2.2) is

*nonhomogeneous*.

An initial value problem for (eq:10.2.2) consists of finding a solution of (eq:10.2.2) that equals a given constant vector at some initial point . We write this initial value problem as

The next theorem gives sufficient conditions for the existence of solutions of initial value problems for (eq:10.2.2). We omit the proof.

- (a)
- Write the system in matrix form and conclude from Theorem thmtype:10.2.1 that every initial value problem for (eq:10.2.3) has a unique solution on .
- (b)
- Verify that is a solution of (eq:10.2.3) for all values of the constants and .
- (c)
- Find the solution of the initial value problem

item:10.2.1b If is given by (eq:10.2.4), then

item:10.2.1c We must choose and in (eq:10.2.4) so that which is equivalent to Solving this system yields , , so is the solution of (eq:10.2.5).

### Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)