1550078481-Ordinary_Differential_Equations__Roberts_

178 Ordinary Differential Equations

1 0 0

W(ex,xex,x^2 ex,0)= 1 1 0 =2/=0.

1 2 2

So the functions ex, xex, and x^2 ex are linearly independent on ( - oo, oo) by
Theorem 4.3.

EXAMPLE 7 Determination of Linear Dependence

or Linear Independence

The functions ex, e-x, and sinh x are solutions on ( - oo, oo) of the third-
order homogeneous linear differential equation

yC3l + yC2) y(ll y = O.

Determine if they are linearly dependent or linearly independent on (-oo, oo ).

SOLUTION

By definition

ex e- x sinhx

W(ex,e- x,sinhx,x) = ex -e-x cosh x

ex e-x sinhx

Evaluating the Wronskia n at x = 0 and computing, we find

1 1 0

W(ex,e-x,sinhx,O) = 1 -1

1 1

1 = 0.

0

So by Theorem 4.3 the functions ex, e-x, and sinh x are linearly dependent
on ( -oo, oo).

The following existence theorem proves that there are at least n linearly
independent solutions on the interval I to the differential equation

where an(x), an-1(x), ... , a1(x), ao(x) are all continuous on the interval I and
an(x) i= 0 for all x E J.