1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 173

THEOREM 4.1 Let the functions y 1 (x),y 2 (x), ... ,ym(x) all be differen-

tiable at least m - 1 times for all x in some interval I. If the functions

Y1, Y2, ... , Ym are linearly dependent on the interval I , then the Wronskian

W(y1, Y2, ... , Ym, x) = 0 for all x E J.

Proof: Since by hypothesis the set of functions {yl ( x), Y2 ( x), ... , Ym ( x)}

is assumed to be linearly dependent on the interval I , there exist constants

c1, c2, ... , Cm not all zero such that

C1Y1(x) + C2Y2(x) + · · · + CmYm(x) = 0 for all x EI.

Differentiating this equation m - 1 times, we find for all x E I

This system of m equations in the m unknowns c 1 , c 2 , ... , Cm may be written

in matrix-vector notation as

(14)

(

~~ ~~

(~-1) (m-1)

Y1 Y2 · · ·

Ym Yr;'. ) ( C1 C2. ) (0) (^0).

......
Ym (m-1) C m^0

(If you are unfamiliar with matrix-vector notation, see section 8.1.) Recall

from linear algebra the theorem which states: "A homogeneous system of

m equation in m unknowns has a nonzero solution if and only if the deter-

minant of the coefficient matrix is zero." Since we have assumed that not

all of the unknowns (the Cis) are zero, equation (14) has a nonzero solution

and, therefore, the determinant of the coefficient matrix is zero- that is,

W(y1,y2, .. ·Ym,x) = 0 for all x EI.

The functions y 1 = sin 2x and y2 = sin x cos x are linearly dependent on the

interval (-00,00), since 1 · y 1 - 2 · Y2 = 1 · sin2x - 2 · sinxcosx = 0 for all

x E (-oo, oo). Hence, by Theorem 4 .1, we must have W(y1, y2, x) = 0 for all

x E ( -oo, oo). Calculating this Wronskian, we find for all x E ( -oo, oo)

sin2x sinx cosx

2 cos 2x - sin^2 x + cos^2 x