1550078481-Ordinary_Differential_Equations__Roberts_

176 Ordinary Differential Equations

THEOREM 4.3 Let an(x), an-i(x), ... , a 1 (x), ao(x) be continuous on

the interval I and let an ( x) -=f. 0 for all x E I. The functions Y1 ( x), Y2 ( x),

... , Yn(x) are linearly independent solutions on the interval I of the n-th

order homogeneous linear differential equation

(15) an(x)y(n)(x) +an-1(x)y(n-l)(x) + · · · +a1(x)y(ll(x) +ao(x)y(x) = 0,

if and only if y 1 (x ), y 2 (x),... , Yn(x) are solutions of (15) on I and the Wron-

skian W(y1, y2, ... , Yn, x) -=f. 0 for some x EI.

Proof: The "if" portion of this theorem is a special case of Theorem 4.2 in

which the functions y 1 (x), y 2 (x), ... , Yn(x) a re all known to be solutions of

the same n-th order homogeneous linear differential equation.

We prove the "only if" portion of this theorem by contradiction. Thus, we

assume that the functions y 1 ( x), Y2 ( x), ... , Yn ( x) are linearly independent

on the interval I and that the Wronskian W (Y1, Y2, ... , Yn, x) = 0 for all

x E I. Choose any a E I. By the theorem from linear algebra stated earlier,

since W(y1, Y2, ... , Yn, a) = 0 there exists a nonzero solution to the following

linear homogeneous system of n equations in the n unknowns c1, c2, ... , Cn

(16)^0

Let the nonzero solution be denoted by k 1 , k 2 , ... , kn and consider the linear

combination

(17)

Since Y1 (x), Y2(x), ... , Yn(x) are solutions of (15) on I , by the superposition

theorem y(x) is a solution of (15) on I. Differentiating (17) n - 1 times

and then evaluating (17) and each derivative at x = a, we find from equa-

tions (16) that y(x) satisfies the conditions

(18) y(a) = 0, y(ll(a) = 0, ... , y(n-l)(a) = 0.

That is, y(x) satisfies the initial value problem consisting of the differen-

tial equation (15) and the initial conditions (18). But by the existence and

uniqueness theorem the unique solution of this initial value problem is the