1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations 367

The development of the theory for systems of linear first-order differential

equations closely parallels that for n-th order linear differential equations. We

state the following theorems without proof.

SUPERPOSITION THEOREM FOR HOMOGENEOUS LINEAR

SYSTEMS

If Y1 and Y2 are any two solutions of the homogeneous linear system

(7) y' = A(x)y, then y3 = C1Y1 + C2Y2 where c1 and c2 are arbitrary scalar

constants is also a solution of (7).

The superposition theorem can easily be generalized to show that if y 1 ,

Y2, ... , Ym are any m solutions of (7) y' = A(x)y, then y = C1Y1 + C2Y2 +

· · ·+CmYm where c 1 , c2, ... , Cm are arbitrary scalar constants is also a solution

of the homogeneous system (7).

EXISTENCE THEOREM FOR HOMOGENEOUS LINEAR

SYSTEMS

If A(x) is continuous on some interval (a,,B)- that is, if aij(x) is a con-

tinuous function on (a, ,B) for all i, j = 1, 2, ... , n, then there exist n linearly

independent solutions of the homogeneous linear system (7) y' = A(x)y on

the interval (a, ,B).

The existence theorem just stated tells us there are n linearly independent

solutions Y1,Y2,... ,yn ofy' = A(x)y on the interval (o:,,B), provided A(x)

is continuous on (o:, ,B).

The following representation theorem tells us how to write every other so-

lution in terms of Y1, Y2, ... , Yn·

REPRESENTATION THEOREM FOR HOMOGENEOUS LINEAR

SYSTEMS

If A(x) is continuous on the interval (o:,,B), ify1,y2, ... , y n are linearly

independent solutions of the homogeneous linear system (7) y' = A(x)y on

(o:, ,B), and if y is any other solution of (7) on (o:, ,B), then there exist scalar

constants C1, c2, ... , Cn such that

y(x) = C1Y1 (x) + C2Y2(x) + · · · + CnYn(x) on (o:, ,B).