1550078481-Ordinary_Differential_Equations__Roberts_

174 Ordinary Differential Equations

= sin 2x( - sin^2 x + cos x^2 ) - 2 cos 2x(sin x cos x)
= (2sinxcosx)(cos2x) - 2sinxcosxcos2x = 0.
From the study of logic, we know that if the statement "A implies B" is
true, then the statement "not B implies not A" is also true. That is, if "A
implies B" is a theorem, then the contrapositive of the statement, which is
"not B implies not A" is also a theorem. The contrapositive of Theorem 4.1
is the following theorem.

THEOREM 4.2 Let the functions Y1(x), Y2(x), ... , Ym(x) all be differen-

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

W(y1,y2, ... ,ym,x) =f. 0 for some x EI, then the functions Y1,y2, ... ,ym

are linearly independent on the interval I.

EXAMPLE 4 Verification of the Linear Independence

of Two Differentiable Functions

Show that ex and e^2 x are linearly independent on the interval (-oo, oo ).

SOLUTION

The functions ex and e^2 x are differentiable on (-oo, oo) and their Wronskian
is

e2x I = 2 e3x _ e3x = e3x _;_ O

2e2x r for all x E (-00,00).

Therefore, by Theorem 4.2, the functions ex and e^2 x are linearly independent
on (-00, 00).

EXAMPLE 5 Verification of the Linear Independence

of Three Differentiable Functions

Show that the functions 1, x, and x^2 are linearly independent on the interval

(-00,00).

SOLUTION

The functions 1 , x, and x^2 are at least twice differentiable on (-oo, oo) and
their Wronskian is

1 x x^2

W(l,x,x^2 ,x) = 0 1 2x = 2 =f. 0 for all x E (-00,00).

0 0 2