1550078481-Ordinary_Differential_Equations__Roberts_

Systems of First-Order Differential Equations 315

In this instance, the independent variable is time, t, the dependent variables

are u1, u2, u3, and U4 and for i = 1,2,3,4 the functions fi(t,u 1 ,u 2 ,u 3 ,u 4 )

are

fi (t, U1, U2, U3, U4) = U2

-k1u1 + k2(u3 - u1)

f2(t,u1,u2,u3,u4) = ----~---'­

m1

f3(t, U1, U2, U3, U4) = U4

-k2(u3 - u1)

f4(t, U1, U2, U3, U4) =.

m2

In chapter 10 we will show how to solve system (4) numerically given the

initial positions and velocities of the two masses.

The first-order systems (2) and (4) are examples of linear systems. Specif-

ically a linear system of first-order equations is one in which each function fi

is linear in the dependent variables.

DEFINITION Linear System of n First-Order Differential

Equations

A linear system of n first-order differential equations has the gen-

eral form

(5)

where aij(x) and bi(x) are all known, real-valued functions of the indepen-

dent variable x.

In system (2), a 11 (x) = -1/30, a 1 2(x) = 0, b1(x) = .5, a21(x) = 1/30,

a22(x) = -1/15, and b2(x) = 0. In system (4), au(x) = ai3(x) = ai4(x) =

b1(x) = 0 and a12(x) = 1; a21(x) = -(k1 + k2)/m1, a22(x) = a24(x) =

b2(x) = 0, and a23(x) = k2/m1; a31(x) = a32(x) = a33(x) = b3(x) = 0

and a3 4 (x) = 1; and a41(x) = k2/m2, a42(x) = a44(x) = b4(x) = 0, and

a 4 3(x) = -k2/m2. Since all aij(x) and all bi(x) in both systems (2) and (4)
are constant, as opposed to variable, these systems are referred to as linear
systems of differential equations with constant coefficients. The following