Page 316 - Advanced engineering mathematics

296 CHAPTER 10 Systems of Linear Differential Equations

Wewill refertothisasa linear system. This system is homogeneous if G(t) is the n × 1
zero matrix, which occurs when each g j (t) is identically zero. Otherwise the system is
nonhomogeneous.
We have an initial value problem for this linear system if the solution is specified at some
value t = t 0 . Here is the fundamental existence/uniqueness theorem for initial value problems

THEOREM 10.1

Let I be an open interval containing t 0 . Suppose A(t) =[a ij (t)] is an n × n matrix of functions
that are continuous on I, and let
⎛ ⎞
g 1 (t)
g 2 (t)
⎜ ⎟
⎜ ⎟
G(t) = ⎜ . ⎟
.
⎝ . ⎠
g n (t)
0
be an n × 1 matrix of functions that are continuous on I.Let X beagiven n × 1 matrix of real
numbers. Then the initial value problem:
X = AX + G;X(t 0 ) = X 0

has a unique solution that is defined for all t in I.

Armed with this result, we will outline a procedure for finding all solutions of the
system (10.1). This will be analogous to the theory of the second order linear differential equa-
tion y + p(x)y +q(x)y = g(x) in Chapter 2. We will then show how to carry out this procedure

to produce solutions in the case that A is a constant matrix.
10.1.1 The Homogeneous System X = AX

If 1 and 2 are solutions of X = AX, then so is any linear combination

c 1 1 + c 2 2 .

This is easily verified by substituting this linear combination into X = AX. This conclusion
extends to any finite sum of solutions.

A set of k solutions X 1 ,··· ,X k is linearly dependent on an open interval I (which can be
the entire real line) if one of these solutions is a linear combination of the others, for all t
in I. This is equivalent to the assertion that there is a linear combination
c 1 X 1 (t) + c 2 X 2 (t) + ··· + c k X k (t) = 0

for all t in I, with at least one of the coefficients c 1 ,··· ,c k nonzero.
We call these solutions linearly independent on I if they are not linearly dependent
on I. This means that no one of the solutions is a linear combination of the others.
Alternatively, these solutions are linearly independent if and only if the only way an
equation
c 1 X 1 (t) + c 2 X 2 (t) + ··· + c k X k (t) = 0
can hold for all t in I is for each coefficient to be zero: c 1 = c 2 = ··· = c k = 0.

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October 14, 2010 20:32 THM/NEIL Page-296 27410_10_ch10_p295-342