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.

Text Source

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