1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 185

The following theorem summarizes the results we obtained in this section.

SUMMARY THEOREM

A. If the functions an(x), an-1(x), ... , a1(x ), and ao(x) and a re all con-

tinuous on the interval I and if an(x) f. 0 for any x in I , then for the

homogeneous n-th order linear differential equation

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

i. there exists a set containing exactly n linearly independent solutions on

t he interval I , say {y 1 (x ), Y2(x ), ... , Yn(x )},

11. the general solution of the homogeneous linear differential equation on I
    is
    Yc(x) = C1Y1(x) + C2Y2(x) + · · · + CnYn(x)

where c 1 , c2, ... , Cn are arbitra ry constants, and

11i. t here exists a unique solution of the homogeneous linear differential

equation which satisfies t he initial conditions

where k 1 , k2, ... , kn are specified constants and xo is some point in J.

B. If the functions an(x), an-I (x ), ... , a 1 (x ), ao(x ) , and b(x) are all contin-

uous on the interval I , if an(x) f. 0 for any x in J , and if b(x) f. 0 for some

x in I, then for the nonhomogeneous n-th order linear differential
equation

an(x)y(n)(x) + an-IY(n-I)(x) + · · · + a1(x)yCil(x) + ao(x)y(x) = b(x)

i. there exists a particular solution of the nonhomogeneous differential e-

quation on I , say yp(x),

ii. the general solution of the nonhomogeneous differential equation on I is

y(x) = Yc(x) + yp(x)

where Yc(x) is the general solution of the associated homogeneous equ a-

tion, and

iii. t here exists a unique solution to nonhomogeneous differential equation
which satisfies the initial conditions

where k 1 , k 2 , ... , kn are specified constants and xo is some point in J.