1550078481-Ordinary_Differential_Equations__Roberts_

368 Ordinary Differential Equations

DEFINITION General Solution of a Homogeneous Linear System

When Y1, y2, ... , Yn are linearly independent solutions of the homoge-

neous linear system (7) y' = A(x)y on the interval (ex, {3), the linear com-

bination y (x) = C1Y1(x) + C2Y2(x) + · · · + CnYn(x) where C1, c2, ... , Cn are

arbitrary scalar constants is called the general solution of (7) on (ex, {3).

Summarizing, the existence theorem says there are at least n linearly inde-

pendent solutions of y' = A(x)y, and the representation theorem states that

there are at most n linearly independent solutions. Thus, the existence theo-

rem gives us license to seek n linearly independent solutions of y' = A(x)y,

and the representation theorem tells us how to write the general solution (all

other solutions) in terms of these solutions. So our task in solving y' = A(x) y

is reduced to one of finding n linearly independent solutions- that is, our
task becomes one of finding a set of n solutions and testing the set for lin-
ear independence. The following theorem aids in the determination of linear
independence of solutions.

THEOREM ON LINEAR INDEPENDENCE OF SOLUTIONS OF

HOMOGENEOUS LINEAR SYSTEMS

Let Y1 (x ), Y2(x ), .. ., Yn(x) be solutions of the homogeneous linear system

(7) y' = A(x) y on some interval (ex, {3). The set offunctions {y 1 , Y2, ... , Yn}

is linearly independent on (ex, {3) if and only if det (y 1 Y2 · · · Yn) '/- 0 for some

xo E (ex, {3).

Consequently, to check a set of n solutions {y 1 (x), Y2(x), ... , Yn(x)} of

y' = A(x)y for linear independence on (ex,{3), we only need to calculate the

determinant of the matrix whose columns are y 1 , Y2, ... , Yn evaluated at some
convenient point xo E (ex,{3). If det (y1(xo) Y2(xo) · · ·Yn(xo)) '/-0, then the
solutions are linearly independent; whereas, if the determinant is zero, then
the solutions are linearly dependent.

EXAMPLE 1 Verifying Linear Independence and Writing the

General Solution

Verify that