1550078481-Ordinary_Differential_Equations__Roberts_

206 Ordinary Differential Equations

Consequently, the set {y 1 ( x), y 2 ( x), y3 ( x)} is linearly independent on the in-
terval ( -oo, oo) and the general solution of the DE (8) on ( -oo, oo) is

where c 1 , c 2 , and c 3 are arbitrary constants.

EXAMPLE 1 Solution of a Homogeneous, Linear Differential

Equation with Constant Coefficients

Find the general solution of the fourth order homogeneous linear differential
equation

(9) y(4) - 2y(2) + y = o.

SOLUTION

The a uxiliary equation corresponding to the DE (9) is

p(r) = r^4 - 2r^2 + 1 = 0.

Factoring, we find
p(r) = (x + 1)^2 (x - 1)^2 = 0.

So the roots of the auxiliary equation are -1, -1, 1, and 1. Hence, t he general
solution of the DE (9) on the interval ( -oo, oo) is

where c 1 , c2, c3, and c 4 are arbitrary constants.

EXAMPLE 2 Solution of a Homogeneous Linear Differential

Equation with Constant Coefficients

Find the general solution of the tenth order homogeneous linear differential
equation.

y(lO) + 3y(^9 ) - 6y(B) - 22y(^7 ) - 3y(^6 ) + 39y(^5 ) + 40y(^4 ) + 12y(^3 ) = 0.

SOLUTION

The auxiliary equation corresponding to this differential equation is

(10) r^10 + 3r^9 - 6r^8 - 22r^7 - 3r^6 + 39r^5 + 40r^4 + 12r^3 = 0.