Systems of First Order Linear Equations

Systems of First Order Linear Equations

(The 2 ? 2 Case)

To Accompany "Elementary Differential Equations" by Boyce and DiPrima

Adam Bowers November 1, 2015

c Adam Bowers 2015 2

Preface

This manuscript is intended to accompany Chapter 7 of the book Elementary Differential Equations by Boyce and DiPrima. The elementary differential equations course at our university does not have linear algebra as a prerequisite, and so it is sometimes felt that the treatment given to first order linear systems in this text is somewhat too general for our audience.

The purpose of this manuscript is to treat the same topic in the same way, but to restrict our attention to the much simpler case of 2 ? 2 matrices.

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Section 1

Introduction

If n is a positive integer, then an n ? n first order system is a list of n first order differential equations having

the form:

dx1 dt

=

F1(x1, . . . , xn, t)

dx2 dt

=

F2(x1, . . . , xn, t)

dx3 dt

=

F3(x1, . . . , xn, t)

...

dxn dt

=

Fn(x1, . . . , xn, t).

The differential system is called a linear system provided that F1, . . . , Fn are linear functions of the variables

x1, . . . , xn.

Note 1.1. The variable t is the independent variable and x1, . . . , xn are dependent variables that are implicitly defined as functions of t.

Example 1.2. The 2 ? 2 linear system

dx1 dt

=

x2

dx2 dt

=

-kx1

- x2

+ cos(t)

describes the motion of a 1 kg mass on a spring acted upon by an external force given by F (t) = cos(t), where t represents time, x1 is the displacement of the mass from equilibrium at time t, x2 is the velocity of the mass at time t, k is the spring constant, and is the damping constant.

Proposition 1.3. An nth order linear differential equation can be transformed into an n ? n linear system. Rather than prove this, we will demonstrate how it can be done with some examples.

Example 1.4. Transform the given differential equation into a first order linear system: y + ty + 3y = sin(t).

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