1550078481-Ordinary_Differential_Equations__Roberts_

172 Ordinary Differential Equations

C1Y1(Xm) + C2Y2(Xm) + · · · + CmYm(Xm) = 0

is C1 = C2 = ... = Cm = 0.

In order to prove that a set of functions { Y1 ( x), Y2 ( x), ... , Ym ( x)} is linearly

dependent on an interval I , we must find explicit constants c1, c2, ... , Cm not
all zero such that c 1 y 1 (x) + c2y2(x) + · · · + CmYm(x) = 0 for all x E J. On the

other hand, to show that a set of functions {y1 (x ), Y2(x ), ... , Ym(x)} is linearly

independent on an interval I, we must prove that we cannot find constants
C1, c2, ... , Cm not all zero such that C1Y1 (x) + c2y2(x) + · · · + CmYm(x) = 0
for all x E J. This is usually very difficult to do directly. However, when

Y1 (x ), Y2 (x ), ... , Ym(x) are all solutions of the same homogeneous linear dif-

ferential equation, then, as we shall soon discover, it is fairly easy to check for
linear dependence or linear independence.

DEFINITION Wronskian

Let the functions y 1 ( x), Y2 ( x), ... , Ym ( x) all be differentiable at least m - 1

times for all x in some interval I. The Wronskian of y 1 , y2, ... , Ym on I is

the determinant

Y1 Y2 Ym

y~ y~ y'rr,

W(y1, y2, ... , Ym, x) =

(m-1)

Y1

(m-1)

Y2 · · ·

(m-1)

Ym-1

The Wronskian is named in honor of Jozef Maria Hoene Wronski (1778-1853),
who was born in Poland, studied mathematics and philosophy in Germany,

and lived much of his life in France. For m = 2 the Wronskian of y 1 and y 2 is

W(y1,y2,x)=I Y7 y~ l=Y1Y~-y~y2.

Y1 Y2

And for m = 3 the Wronskian of Y1, Y2, and y3 is

YI Y2 Y3

W (y1, Y2, Y3, x) = Y~ Y~ Y~

y~ y~ y~