1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 171

DEFINITIONS Linearly Dependent and Linearly Independent

Let {YI ( x), Y2 ( x), ... , Ym ( x)} be a set of functions defined on an interval

I.

The set is linearly dependent on the interval I if there exist constants

CI, c2, ... , Cm not all zero such that

for all x EI.

Otherwise, the set is linearly independent on the interval I.

That is, the set of functions {YI (x ), y2(x ), ... , Ym(x)} is linearly indepen-

dent on the interval I if and only if CIYI(x) + C2Y2(x) + · · · + CmYm(x) = 0

for all x in I implies CI = c 2 = · · · = Cm = 0. In other words, the set of

functions is linearly independent on the interval I if and only if the only way

the linear combination CIYI (x) + c2y2(x) + · · · + CmYm(x) can be identically

zero on the interval I is for CI = c2 = · · · = Cm = 0.

For example, the following sets of functions are linearly independent on the

interval I= (-oo, oo )- that is, on the entire real line: {1, x, 3x^2 , x^5 , -6xI^0 } ,

{e^2 x xe^2 x e^3 x 5x} and {sin x cos 3x e-x xex sin 2x}

' ' ' ' ' ' '.

The set { I_ 2 , -

1

-, -

1

-} is not linearly independent on (-oo, oo ). This

x x-l x+l

set is linearly independent on each of the intervals (-oo, -1), (-1, 0), (0, 1),

and (1, oo). The set is not linearly independent on any interval which contains

-1, 0, or 1, since some function in the set is not defined at -1, 0, or l.

The set {2, x, 3 - 4x} is linearly dependent on every interval I, since

( -3) · 2 + (8) · x + (2) · (3 - 4x) = 0 for all real x.

The set {x, lxl} is linearly dependent on the interval (0, 3), since

(-1) · x + (1) · lx l = 0 for all x in (0,3). But the set {x, lxl} is linearly

independent on the interval (-3, 3), since CIX + c2lx l = 0 for x = 1 implies

CI + c2 = 0 and since CIX + c2lxl = 0 for x = -1 implies -CI+ c2 = 0.

The simultaneous solution of CI + c2 = 0 and -CI + c2 = 0 is CI = c2 = 0.

This example illustrates that the set of functions {YI(x),y2(x), ... ,ym(x)}

can be shown to be linearly independent on an interval I by evaluating the

linear combination CI YI ( x) + c2 Y2 ( x) + · · · +Cm Ym ( x) = 0 at m distinct points

XI, x 2 , ... , Xm in the interval I and showing that the only solution of them
simultaneous equations