Chapter 6:
Systems of Linear
Differential Equations
Example 10.1
The 2×2 system
x1  3x1  3x2  8
x2  x1  5x2  4e3t
Chapter 10,P362
THEOREM 10.1
Existence and Uniqueness


Let I be an open interval containing t 0 .
Suppose each aij t  and g j t  are continuous on
I. Let X 0 be a given n × 1 matrix of real
numbers. Then the initial value problem
X   AX  G
X t0   X 0
has a unique solution defined for all t in I.
Chapter 10,P364
Example 10.2
Consider the initial value problem
x1  x1  tx2  cost 
x2  t 3 x1  et x2  1  t
x1 0  2
x2 0  5
Chapter 10,P364
THEOREM 10.2

Let 1..... k be solutions of X   AX , all
defined on some open interval I. Let c1......ck
be any real numbers. Then the linear
combination c11  .....ck  k is also a
solution of X   AX , defined on I.
Chapter 10,P364
DEFINITION 10.1

Linear Dependence
Solutions 1, 2 .......k of X   AX , defined on
an interval I, are linearly dependent on I if
one solution is a linear combination of the
others on this interval.
Linear Independence
Solutions 1, 2 .......k of X   AX , defined on an
interval I, are linearly independent on I if no
solution in this list is a linear combination of
the others on this interval.
Chapter 10,P365
Example 10.3
Consider the system
Chapter 10,P366
THEOREM 10.3
Test for Linear Independence of
Solutions

Suppose that
are solutions of X   AX on an open interval I.
Chapter 10,P367



Let t 0 be any number in I. Then
1. 1,  2 ...... n are linearly independent on I if
and only if 1 t0 ,..... n t0  are linearly
independent, when considered as vectors in R n .
2. 1,  2 ...... n are linearly independent on I if
and only if
Chapter 10,P367
Example 10.4
From the preceding example,
  2e3t 
1 t    3t 
 e 
and
 1  2t e3t 

 2 t   
3t

te


Chapter 10,P367
THEOREM 10.4
Let A =[ aij t  ] be an n × n matrix of
functions that are continuous on an open
interval I. Then
1. The system X   AX has n linearly
independent solutions defined on I.
2. Given any n linearly independent solutions
1,..... n defined on I, every solution on
I is a linear combination of 1 ,..... n .

Chapter 10,P369
Example 10.5
Previously we saw that
  2e3t 
1 t    3t 
 e 
 1  2t e3t 
and  2 t    te3t 


are linearly independent solutions of
Chapter 10,P370
DEFINITION 10.2

Ω is a fundamental matrix for the n × n
system X   AX if the columns of Ω are
linearly independent solutions of this
system.
Chapter 10,P371
Example 10.6
Solve the initial value problem
1  4 

X   
1 5 
  2
X 0   
 3 
Chapter 10,P372
THEOREM 10.5
Let Ω be a fundamental matrix for X   AX
,
and let  p be any solution of X   AX  G .
Then the general solution of X   AX  G
is X  C  p , in which C is an n×1 matrix
of arbitrary constants.

Chapter 10,P372
THEOREM 10.6

t

e
Let A be an n × n matrix of real numbers. Then
is a nontrivial solution of X   AX if and only if
λis an eigenvalue of A, with associated
eigenvector
.

Chapter 10,P374
THEOREM 10.7

Let A be an n × n matrix of real numbers.
Suppose A has eigenvalues 1......n , and
suppose there are associated eigenvectors
1..... n that are linearly independent.
Then 1e1t ,.... n ent are linearly
independent solutions of X   AX , on the
entire real line.
Chapter 10,P374
Example 10.7
Consider the system
Chapter 10,P374
Example 10.8

Solve the system
Chapter 10,P375
Example 10.9
A Mixing Problem


Two tanks are connected by a series of pipes, as
shown in Figure 10.1. Tank 1 initially contains 20
liters of water in which 150 grams of chlorine are
dissolved. Tank 2 initially contains 50 grams of
chlorine dissolved in 10 liters of water.
Beginning at time t = 0, pure water is pumped
into tank 1 at a rate of 3 liters per minute ,while
chlorine/water solutions are interchanged
between the tanks and also flow out of both
tanks at the rates shown. The problem is to
determine the amount of chlorine in each tank at
Chapter 10,P375
any time t > 0

At the given rates of input and discharge of
solutions, the amount of solution in each tank
will remain constant. Therefore, the ratio of
chlorine to chlorine/water solution in each
tank should, in the long run, approach that of
the input, which is pure water. We will use
this observation as a check of the analysis we
are about to do.
Chapter 10,P376
THEOREM 10.8

Let A be an n × n real matrix. Let α+iβ be a
complex eigenvalue with corresponding
eigenvector U + iV, in which U and V are real n
× 1 matrices. Then
et [U cost   V sin t ]
et [U sin t   V cost ]

are real linearly independent solutions of X   AX
Chapter 10,P378
Example 10.10
Solve the system with X   AX
Chapter 10,P378
Example 10.11

We will solve the
system X   AX
, with
Chapter 10,P380
Example 10.12
Consider the system X   AX , in which
Chapter 10,P381
Example 10.13
Solve
Chapter 10,P386
DEFINITION 10.3
Exponential Matrix

The exponential matrix e At is the n × n
matrix defined by
1 22 1 33
e  I n  At  A t  A t  ....
2!
3!
At
Chapter 10,P387
THEOREM 10.9

Let B be an n × n real matrix. Suppose
AB = BA. Then
e
 A B t
e e
At Bt
Chapter 10,P387
LEMMA 10.1

For any real n×1 constant matrix K, e At K
is a solution of
.



t   e K  Ae K  At 
At
At
Chapter 10,P387
THEOREM 10.10

e
At
is a fundamental matrix for
X   AX
Chapter 10,P388
LEMMA 10.2

Let A be an n × n real matrix and K an n × 1
real matrix. Let μ be any number. Then
I nt
t
1. e K  e K
2. e At K  e t e AI n t K
Chapter 10,P388
Example 10.14
Consider X   AX , where
Chapter 10,P390
Example 10.15
Solve the system
Chapter 10,P395
Example 10.16
Consider the system
Chapter 10,P397
Example 10.17
Consider
Chapter 10,P399