1550078481-Ordinary_Differential_Equations__Roberts_

170 Ordinary Differential Equations

Since y 1 (x) is a solution of (12) on I ,

a3(x)y~' + a2(x)y~ + a 1 (x)y~ + ao(x)y 1 = 0 for all x EI

and since y2(x) is a solution of (12) on I,

Differentiating y(x) = c 1 y 1 (x) + C2Y2(x) three times, we find

and

y' (x) = C1Y~ (x) + c2y~(x),

y" ( X) = C1 y~ ( X) + C2 Y~ ( X) ,

y'"(x) = c1y~^1 (x) + c2y~^1 (x).

Substituting into the left-hand side of (12) for y, y', y", and y'", we see that

a3(x)y^111 + a2(x)y" + a1(x)y^1 + ao(x)y

= a3(x)[c1y~^1 (x) + c2y~^1 (x)] + a2(x)[c1y~ (x) + c2y~ (x)]

+ a1 (x) [c1y~ (x) + c2y~ (x )] + ao(x) [c1Y1 (x) + C2Y2 (x )]

= c1[a3(x)y~^1 + a2(x )y~ + a1 (x)y~ + ao(x)y1]

+ c2[a3(x)y~^1 + a2(x)y~ + a1(x)y~ + ao(x)y2]

=c10+c20=0 forallxEJ.

For example, y 1 (x) = e^2 x and y 2 (x) = sinx are both solutions of the third-
order homogeneous li near differential equation

(13) y'" - 2y" + y' - 2y = 0

on the interval (-oo, oo ). Consequently, y(x) = c 1 e^2 x + c 2 sinx is a solution
to (13) on (-00,00). In particular, z(x) = 3e^2 x - 5sinx is a solution of (13)
on (-00,00).

The superposition theorem states that any linear combination of solutions
of an n-th order homogeneous linear differential equation is a solution of the
same differential equation. A useful corollary of the superposition theorem
is the following: If y(x) is any solution of an n-th order homogeneous linear
differential equation, then cy(x) is also a solution for ·any arbitrary constant
c. The superposition theorem appli es only applies to homogeneous linear dif-
ferential equations. It does not apply to nonhomogeneous linear equations or
nonlinear equations.