1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 169

Since Y2, y~, and Y2 are defined on ( - oo, oo ), Y2 = ex is a solution of the

DE (10) on ( - oo, oo).

b. Differentiating the linear combination y = c1y 1 + C2Y2 = c 1 e-^2 x + c 2 ex

twice, yields

and

y II = C1Y1 II + C2Y2 II =^4 C1e -2x + C2e X.

Substituting into (10), we find

yll + y' - 2y

= (4c1e-^2 x + C2ex) + (-2c1e-^2 x + C2ex) - 2(c1e-^2 x + c2ex)

= c1(4 - 2 - 2)e-^2 x + c2(l + 1-2)ex

= oe-^2 x + Oex = 0.

Since Y1, y~, y{, Y2, y~, and v2 are defined and continuous on ( - oo, oo),

y(x) = C1Y1(x) + c2y2(x) = c1e-^2 x + c2ex is defined and continuous on

(-00,00). And since the linear combination y(x) = c 1 y 1 (x) + c2y2(x)

satisfies the DE (10) on (-oo, oo) for arbitrary constants c 1 and c 2 , y(x)

is a solution of (10) on ( -oo, oo).

The following superposition theorem generalizes this result for homogeneous
linear differential equations.

SUPERPOSITION THEOREM

If y 1 ( x), Y2 ( x), ... , Yk ( x) are solutions on the interval I of the homogeneous

linear differential equation

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

then the linear combination y(x) = C1Y1(x) + C2Y2(x) + · · · + CkYk(x) where

c 1 , c 2 , ... , Ck are arbitrary constants is a solution of (11) on I.

Proof: We will prove this theorem for the special case k = 2 and n = 3.

Then it will be obvious how to prove the theorem in general. We as-

sume that y 1 (x) and y 2 (x) are solutions on an interval I of the third-order,

homogeneous linear differential equation

(12) a3(x)y^111 + a2(x)y^11 + ai(x)y' + ao(x)y = 0.