1550078481-Ordinary_Differential_Equations__Roberts_

Systems of First-Order Differential Equations 317

(7)

where a, b, c, and dare positive constants. The system (7) is nonlinear due to

the terms -by1y2 and dy1y2.

DEFINITION Solution of a System of n First-Order Differential

Equations

A system of n first-order differential equations

y~ =Ji (x, Y1, Y2, · · ·, Yn)

(1) Y~ = f2(x, Y1, Y2, · · ·, Yn)

h as a solution on the interval (a, /3), if there exists a set of n functions

{y1(x), Y2(x),... , Yn(x)} which all have continuous first derivatives on the

interval I and which satisfy (1) on I. The set of functions

is call ed a solution of system (1) on the interval I.

For example, the set of two functions { Y1 ( x) = sin x, Y2 ( x) = cos x} is a
solution of the system of two, first-order, linear differential equ ations

Y~ = Y2 = fi(x, y1,Y2)

(8)

Y~ = - Yi = f2(x, Y1, Y2)

on the interval (-00,00). (You should verify this fact by showing that the
derivatives ofy 1 (x) = sinx and Y2(x) = cosx are both continuous on the in-

terval!= (-oo, oo) and that y~ (x) = Y2(x) and y~(x) = -y1 (x) for all x E J.)

The set of two functions {z 1 (x) = Asinx +B cosx, z2(x) = Acosx - B sinx}

where A and B are arbitrary constants is also a solution to the system (8)
on the interval (-oo, oo ). Verify this fact. The solution { z 1 (x), z2(x)} of the
system (8) is the general solution of (8). In section 8.3, we will show how to