20.2: 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 Equation can be written in matrix form as

or more briefly as

where

We call the coefficient matrix of Equation and the forcing function. We’ll say that and are continuous if their entries are continuous. If , then Equation is homogeneous; otherwise, Equation is nonhomogeneous.

An initial value problem for Equation consists of finding a solution of Equation 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 Equation . We omit the proof.

Theorem 20.2.1 : Existence

Suppose the coefficient matrix and the forcing function are continuous on , let be in , and let be an arbitrary constant -vector. Then the initial value problem

has a unique solution on .

Example 20.2.1

1. Write the system in matrix form and conclude from Theorem 20.2.1 that every initial value problem for Equation has a unique solution on .
2. Verify that is a solution of Equation for all values of the constants and .
3. Find the solution of the initial value problem

Solution a

The system Equation can be written in matrix form as

An initial value problem for Equation can be written as

Since the coefficient matrix and the forcing function are both continuous on , Theorem 20.2.1 implies that this problem has a unique solution on .

Solution b

If is given by Equation , then

Solution c

We must choose and in Equation so that

which is equivalent to

Solving this system yields , , so

is the solution of Equation .

Note

The theory of linear systems of differential equations is analogous to the theory of the scalar n-th order equation as developed in Sections 9.1. For example by rewriting Equation as an equivalent linear system it can be shown that Theorem 20.2.1 implies Theorem 9.1.1 (Exercise 10.2.12).