1550078481-Ordinary_Differential_Equations__Roberts_

370 Ordinary Differential Equations

To determine whether the set {y 1 (x),y2(x)} is linearly dependent or lin-
early independent on (0, oo ), we compute det (y 1 (x) Y2(x)) at some convenient
point x 0 E ( 0, oo). As a matter of convenience, we decided to choose xo = 1.
Computing, we get

det (y 1 (1) y2(l)) = det G D = 1 - 2 = -1.

Since det (y 1 (1) y 2 (1)) = -1-/=-0, the vectors Y1 and Y2 are linearly indepen-

dent on (0, oo). Therefore, by the representation theorem, the general solution
of (8) on (0, oo) is

where c 1 and c 2 are arbitrary scalar constants.

Now let us consider the nonhomogeneous linear system of equations

(9) y' = A(x)y + b(x)

where y, y', and b(x) are n x 1 column vectors; A(x) is an n x n matrix; and
b(x)-/=-0.

DEFINITIONS Associated Homogeneous System and Particular

Solution

The system of differential equations

(10) y' = A(x)y

is called the associated homogeneous system for the nonhomogeneous

system (9) y' = A(x)y + b(x).

Any solution Yp(x) of the nonhomogeneous system (9) which includes no

arbitrary constant is called a particular solution of (9).

The following theorem tells us how to write the general solution of the
nonhomogeneous linear system ( 9).