1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations

REPRESENTATION THEOREM FOR NONHOMOGENEOUS

LINEAR SYSTEMS

371

If A ( x) is continuous on the interval (a , {3), if y P ( x ) is any particular sol u-

tion on (a, {3) of the nonhomogeneous linear system (9) y' = A(x)y + b(x),

and ify1, Y2, ... , y n are n linearly independent solutions on the interval

(a,{3) of the associated homogeneous linear system (10) y' = A(x)y , then

every solution of the nonhomogeneous system (9) on (a, {3) has the form

(11) y(x) = C1Y1 + C2Y2 + · · · + CnY n + Y p

where c 1 , c2, ... , Cn are scalar constants.

Since every solution of (9) can be written in the form of equation (11),
this equation is called the general solution of the nonhomogeneous sys-
tem (9). The general solution of the associated homogeneous system, namely
Ye= C1Y1 +c2Y2 + · · · +cnYn, is called the complementary solution. Thus,

the general solution of the nonhomogeneous system (9) is y =Ye + Y p where

y e is the complementary solution and Y p is any particular solution of the non-
homogeneous system. So to find the general solution of the nonhomogeneous
system (9) y' = A(x)y+ b(x), we first find the general solution of the associ-
ated homogeneous system (10) y ' = A(x)y , next we find a particular solution
of the nonhomogeneous system (9), and then we add the results.

EXAMPLE 2 Verifying a Particular Solution and Writing the

General Solution

Verify that

is a particular solution on the interval (0 , oo) of the nonhomogeneous system

(12)

(

2 -1)

y' ~ : :' y + (.: ) ~ A(x) y +b(x)

and write the general solut ion of (12) on (O,oo).