1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 175

Therefore, by Theorem 4.2, the functions 1, x, and x^2 are linearly independent

on (-00,00).

If the Wronskian of a set of functions is zero at every point in an interval,

that does not imply the set of functions is linearly dependent on the interval.

Consider the functions y 1 ( x) = x^3 and y2 ( x) = Ix I 3 on the interval ( -2, 2).

Differentiating, we find y~ ( x) = 3x^2 and

x<O

x=O

x>O

(Verify that Y2 (0) exists and has the value zero.) Computing the Wronskian

of { x^3 , lx l^3 } on (-2, 0), we find

x3 -x3

W(x^3 , lxl^3 , x) = = -3x^5 + 3x^5 = 0.

3x^2 -3x^2

At 0, we get

0 0

W(x^3 , lxl^3 , 0) = = 0.

0 0

And on (0, 2), we see

x3 x3

W(x^3 , lx l^3 , x) = = 3x^5 - 3x^5 = 0.

3x^2 3x^2

Hence, W(x^3 , lxl^3 , x) = 0 for every x E (-2, 2). Now assume there exist
constants c1 and c2 such that c1 Y1 ( x) + c2y2 ( x) = c1 x^3 + c2 Ix I 3 = 0 for all
x E (-2, 2). For x = -1, we must have -c 1 + c 2 = 0 and for x = 1, we must

have c 1 + c 2 = 0. Simultaneously solving these two equations in c 1 and c 2 , we

find c 1 = c2 = 0, which shows that the set { x^3 , lxl^3 } is linearly independent

on (-2, 2). Hence, {x^3 , lxl^3 } is a set of functions which is linearly independent

on (-2, 2) and whose Wronskian is identically zero on (-2, 2).

We now prove the following important theorem regarding the relationship

of linearly independent solutions of n-th order homogeneous linear differential

equations and the Wronskian of the solutions.