Chapter 7

Systems of Linear Differential Equations

Most engineering devices have more than one moving part. Two masses joined by a spring, the currents in a network with several loops, and salt passing between connected tanks are all described by several unknown functions whose rates of change depend on each other. Such problems lead to systems of differential equations. In this chapter we study linear systems, mainly with constant coefficients, and you will see that the ideas of Chapter 4 carry over almost unchanged.

The chapter has two halves. The first works with two unknowns $x (t)$ , $y (t)$ and needs no matrices:

7.1 Differential operators and the operator method: calculating with $D = d / d t$ , and eliminating one unknown to solve a system.
7.2 Applications: coupled mass–spring systems, two-loop electric networks and interconnected tanks.
7.3 Basic theory: the normal form $x^{'} = a_{11} x + a_{12} y + F_{1}$ , $y^{'} = a_{21} x + a_{22} y + F_{2}$ , existence and uniqueness, the Wronskian and fundamental sets.
7.4 Homogeneous constant-coefficient systems: exponential solutions, the characteristic equation and its three cases, and a first look at the phase plane.

The second half (Sections 7.5–7.8) rewrites everything with matrices and vectors, which handles any number of unknowns: eigenvalues and eigenvectors, the matrix method for two and for $n$ equations, and the general theorems behind it.

What you will be able to do

Solve constant-coefficient linear systems by elimination and know how many arbitrary constants to expect.
Set up systems of equations for mechanical, electrical and mixing problems.
Decide whether two solutions of a system form a fundamental set and solve initial-value problems.
Solve $2 \times 2$ homogeneous systems in all cases and sketch and classify their phase portraits.

Prerequisites

You need constant-coefficient linear equations (Section 4.2) and undetermined coefficients (Section 4.3), the basic theory of Section 4.1, complex numbers and Euler's formula, and $2 \times 2$ determinants. The applications use the mass–spring and circuit models of Chapter 5 and the mixing problems of Section 3.3.