System of First Order ODEs - 1 | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

 Page 1


CHAPTER 1
System of First Order Di?erential Equations
Inthischapter,wewilldiscusssystemof?rstorderdi?erentialequa-
tions. Therearemanyapplicationsthatinvolving?ndseveralunknown
functions simultaneously . Those unknown functions are related by a
set of equations that involving the unknown functions and their ?rst
derivatives. For example, in Chapter Two, we studied the epidemic of
contagious diseases. Now if
² S(t)denotesnumberofpeoplethatissusceptibletothedisease
but not infected yet.
² I(t) denotes number of people actually infected.
² R(t) denotes the number of people have recovered.
If we assume
² The fraction of the susceptible who becomes infected per unit
time is proportional to the number infected, b is the propor-
tional number.
² A?xedfractionrS oftheinfectedpopulationrecoversperunit
time, 0·r· 1.
² A ?xed fraction of the recovers g become susceptible and in-
fected, 0·g· 1. proportional function.
The system of di?erential equations model this phenomena are
S
0
=¡bIS +gR
I
0
=bIS¡rI
R
0
=rI¡gR
The numbers of unknown function in a system of di?erential equa-
tions can be arbitrarily large, but we will concentrate ourselves on 2 to
3 unknown functions.
1. Principle of superposition
Let a
ij
(t); b
j
(t) i = 1;2;¢¢¢ ;n and j = 1;2;¢¢¢ ;n be known
function, and x
i
t; i =1;2;¢¢¢ ;nbeunknownfunctions, thelinear?rst
Page 2


CHAPTER 1
System of First Order Di?erential Equations
Inthischapter,wewilldiscusssystemof?rstorderdi?erentialequa-
tions. Therearemanyapplicationsthatinvolving?ndseveralunknown
functions simultaneously . Those unknown functions are related by a
set of equations that involving the unknown functions and their ?rst
derivatives. For example, in Chapter Two, we studied the epidemic of
contagious diseases. Now if
² S(t)denotesnumberofpeoplethatissusceptibletothedisease
but not infected yet.
² I(t) denotes number of people actually infected.
² R(t) denotes the number of people have recovered.
If we assume
² The fraction of the susceptible who becomes infected per unit
time is proportional to the number infected, b is the propor-
tional number.
² A?xedfractionrS oftheinfectedpopulationrecoversperunit
time, 0·r· 1.
² A ?xed fraction of the recovers g become susceptible and in-
fected, 0·g· 1. proportional function.
The system of di?erential equations model this phenomena are
S
0
=¡bIS +gR
I
0
=bIS¡rI
R
0
=rI¡gR
The numbers of unknown function in a system of di?erential equa-
tions can be arbitrarily large, but we will concentrate ourselves on 2 to
3 unknown functions.
1. Principle of superposition
Let a
ij
(t); b
j
(t) i = 1;2;¢¢¢ ;n and j = 1;2;¢¢¢ ;n be known
function, and x
i
t; i =1;2;¢¢¢ ;nbeunknownfunctions, thelinear?rst
order system of di?erential equation for x
i
(t) is the following,
x
0
1
(t) = a
11
(t)x
1
(t)+a
12
(t)x
2
(t)+¢¢¢+a
1n
(t)x
n
(t)+b
1
(t)
x
0
2
(t) = a
21
(t)x
1
(t)+a
22
(t)x
2
(t)+¢¢¢+a
2n
(t)x
n
(t)+b
2
(t)
x
0
3
(t) = a
31
(t)x
1
(t)+a
32
(t)x
2
(t)+¢¢¢+a
3n
(t)x
n
(t)+b
3
(t)
.
.
.
x
0
n
(t) = a
n1
(t)x
1
(t)+a
n2
(t)x
2
(t)+¢¢¢+a
nn
(t)x
n
(t)+f
1
(t)
Let x(t) be the column vector of unknown functions x
i
t; i =
1;2;¢¢¢ ;n, A(t) = (a
ij
(t); and b(t) be the column vector of known
functions b
i
t; i = 1;2;¢¢¢ ;n; we can write the ?rst order system of
equations as
x
0
(t)=A(t)x(t)+b(t) (1)
² When n =2, the linear ?rst order system of equations for two
unknown functions in matrix form is,
·
x
0
1
(t)
x
0
2
(t)
¸
=
·
a
11
(t) a
12
(t)
a
21
(t) a
22
(t)
¸·
x
1
(t)
x
2
(t)
¸
+
·
b
1
(t)
b
2
(t)
¸
² When n = 3, the linear ?rst order system of equations for
three unknown functions in matrix form is,
2
4
x
0
1
(t)
x
0
2
(t)
x
0
3
(t)
3
5
=
2
4
a
11
(t) a
12
(t) a
13
a
21
(t) a
22
(t) a
23
a
31
(t) a
32
(t) a
33
3
5
2
4
x
1
(t)
x
2
(t)
x
3
t
3
5
+
2
4
b
1
(t)
b
2
(t)
b
3
(t)
3
5
Asolution of equation (1) on the open interval I is a column vec-
tor function x(t) whose derivative (as a vector-values function) equals
A(t)x(t)+b(t): The following theorem gives existence and uniqueness
of solutions,
Theorem 1.1. Ifthevector-valuedfunctionsA(t)andb(t)arecon-
tinuous over an open interval I contains t
0
; then the initial value prob-
lem
½
x
0
(t)=A(t)x(t)+b(t)
x(t
0
)=x
0
has an unique vector-values solution x(t) that is de?ned on entire in-
terval I for any given initial value x
0
:
Whenb(t)´0; the linear ?rst order system of equations becomes
x
0
(t)=A(t)x(t);
which is called a homogeneous equation.
As in the case of one equation, we want to ?nd out the general
solutions for the linear ?rst order system of equations. To this end, we
?rst have the following results for the homogeneous equation,
Page 3


CHAPTER 1
System of First Order Di?erential Equations
Inthischapter,wewilldiscusssystemof?rstorderdi?erentialequa-
tions. Therearemanyapplicationsthatinvolving?ndseveralunknown
functions simultaneously . Those unknown functions are related by a
set of equations that involving the unknown functions and their ?rst
derivatives. For example, in Chapter Two, we studied the epidemic of
contagious diseases. Now if
² S(t)denotesnumberofpeoplethatissusceptibletothedisease
but not infected yet.
² I(t) denotes number of people actually infected.
² R(t) denotes the number of people have recovered.
If we assume
² The fraction of the susceptible who becomes infected per unit
time is proportional to the number infected, b is the propor-
tional number.
² A?xedfractionrS oftheinfectedpopulationrecoversperunit
time, 0·r· 1.
² A ?xed fraction of the recovers g become susceptible and in-
fected, 0·g· 1. proportional function.
The system of di?erential equations model this phenomena are
S
0
=¡bIS +gR
I
0
=bIS¡rI
R
0
=rI¡gR
The numbers of unknown function in a system of di?erential equa-
tions can be arbitrarily large, but we will concentrate ourselves on 2 to
3 unknown functions.
1. Principle of superposition
Let a
ij
(t); b
j
(t) i = 1;2;¢¢¢ ;n and j = 1;2;¢¢¢ ;n be known
function, and x
i
t; i =1;2;¢¢¢ ;nbeunknownfunctions, thelinear?rst
order system of di?erential equation for x
i
(t) is the following,
x
0
1
(t) = a
11
(t)x
1
(t)+a
12
(t)x
2
(t)+¢¢¢+a
1n
(t)x
n
(t)+b
1
(t)
x
0
2
(t) = a
21
(t)x
1
(t)+a
22
(t)x
2
(t)+¢¢¢+a
2n
(t)x
n
(t)+b
2
(t)
x
0
3
(t) = a
31
(t)x
1
(t)+a
32
(t)x
2
(t)+¢¢¢+a
3n
(t)x
n
(t)+b
3
(t)
.
.
.
x
0
n
(t) = a
n1
(t)x
1
(t)+a
n2
(t)x
2
(t)+¢¢¢+a
nn
(t)x
n
(t)+f
1
(t)
Let x(t) be the column vector of unknown functions x
i
t; i =
1;2;¢¢¢ ;n, A(t) = (a
ij
(t); and b(t) be the column vector of known
functions b
i
t; i = 1;2;¢¢¢ ;n; we can write the ?rst order system of
equations as
x
0
(t)=A(t)x(t)+b(t) (1)
² When n =2, the linear ?rst order system of equations for two
unknown functions in matrix form is,
·
x
0
1
(t)
x
0
2
(t)
¸
=
·
a
11
(t) a
12
(t)
a
21
(t) a
22
(t)
¸·
x
1
(t)
x
2
(t)
¸
+
·
b
1
(t)
b
2
(t)
¸
² When n = 3, the linear ?rst order system of equations for
three unknown functions in matrix form is,
2
4
x
0
1
(t)
x
0
2
(t)
x
0
3
(t)
3
5
=
2
4
a
11
(t) a
12
(t) a
13
a
21
(t) a
22
(t) a
23
a
31
(t) a
32
(t) a
33
3
5
2
4
x
1
(t)
x
2
(t)
x
3
t
3
5
+
2
4
b
1
(t)
b
2
(t)
b
3
(t)
3
5
Asolution of equation (1) on the open interval I is a column vec-
tor function x(t) whose derivative (as a vector-values function) equals
A(t)x(t)+b(t): The following theorem gives existence and uniqueness
of solutions,
Theorem 1.1. Ifthevector-valuedfunctionsA(t)andb(t)arecon-
tinuous over an open interval I contains t
0
; then the initial value prob-
lem
½
x
0
(t)=A(t)x(t)+b(t)
x(t
0
)=x
0
has an unique vector-values solution x(t) that is de?ned on entire in-
terval I for any given initial value x
0
:
Whenb(t)´0; the linear ?rst order system of equations becomes
x
0
(t)=A(t)x(t);
which is called a homogeneous equation.
As in the case of one equation, we want to ?nd out the general
solutions for the linear ?rst order system of equations. To this end, we
?rst have the following results for the homogeneous equation,
Theorem1.2. PrincipleofSuperpositionLetx
1
(t); bx
2
(t);¢¢¢ ;x
n
(t)
be n solutions of the homogeneous linear equation
x
0
(t)=A(t)x(t)
on the open interval I: If c
1
; c
2
; ¢¢¢ ; c
n
are n constants, then the
linear combination
c
1
x
1
(t)+c
2
x
2
(t)+c
3
x
3
(t)+¢¢¢+c
n
x
n
(t)
is also a solution on I:
Example 1.1. Let
x
0
(t)=
·
1 0
0 ¡2
¸
x(t)
, x
1
(t)=
·
e
t
0
¸
and x
2
(t)=
·
0
e
¡2t
¸
are two solutions, as
bx
0
1
(t)=
·
(e
t
)
0
0
¸
=
·
e
t
0
¸
=
·
1 0
0 ¡2
¸·
e
t
0
¸
and
bx
0
2
(t)=
·
0
(e
¡2t
)
0
¸
=
·
0
¡2e
¡2t
¸
=
·
1 0
0 ¡2
¸·
0
e
¡2t
¸
By the Principle of Superposition, for any two constants c
1
and c
2
x(t)=c
1
x
1
(t)+c
2
x
2
(t)=c
1
·
e
t
0
¸
+c
2
·
0
e
¡2t
¸
=
·
c
1
e
t
c
2
e
¡2t
¸
is also solution. We shall see that it is actually the general solution.
The next theorem gives the general solution of linear system of
equations,
Theorem 1.3.
- Let x
1
(t); x
2
(t); ¢¢¢ ; bx
n
(t) be n linearly independent (as vectors)
solution of the homogeneous system
x
0
(t)=A(t)x(t);
then for any solutionx
c
(t) there exists n constants c
1
; c
2
; ¢¢¢ ; c
n
such
that
x
c
(t)=c
1
x
1
(t)+c
2
x
2
(t)+¢¢¢+c
n
x
n
(t):
We call x
c
(t) the general solution of the homogeneous system.
Page 4


CHAPTER 1
System of First Order Di?erential Equations
Inthischapter,wewilldiscusssystemof?rstorderdi?erentialequa-
tions. Therearemanyapplicationsthatinvolving?ndseveralunknown
functions simultaneously . Those unknown functions are related by a
set of equations that involving the unknown functions and their ?rst
derivatives. For example, in Chapter Two, we studied the epidemic of
contagious diseases. Now if
² S(t)denotesnumberofpeoplethatissusceptibletothedisease
but not infected yet.
² I(t) denotes number of people actually infected.
² R(t) denotes the number of people have recovered.
If we assume
² The fraction of the susceptible who becomes infected per unit
time is proportional to the number infected, b is the propor-
tional number.
² A?xedfractionrS oftheinfectedpopulationrecoversperunit
time, 0·r· 1.
² A ?xed fraction of the recovers g become susceptible and in-
fected, 0·g· 1. proportional function.
The system of di?erential equations model this phenomena are
S
0
=¡bIS +gR
I
0
=bIS¡rI
R
0
=rI¡gR
The numbers of unknown function in a system of di?erential equa-
tions can be arbitrarily large, but we will concentrate ourselves on 2 to
3 unknown functions.
1. Principle of superposition
Let a
ij
(t); b
j
(t) i = 1;2;¢¢¢ ;n and j = 1;2;¢¢¢ ;n be known
function, and x
i
t; i =1;2;¢¢¢ ;nbeunknownfunctions, thelinear?rst
order system of di?erential equation for x
i
(t) is the following,
x
0
1
(t) = a
11
(t)x
1
(t)+a
12
(t)x
2
(t)+¢¢¢+a
1n
(t)x
n
(t)+b
1
(t)
x
0
2
(t) = a
21
(t)x
1
(t)+a
22
(t)x
2
(t)+¢¢¢+a
2n
(t)x
n
(t)+b
2
(t)
x
0
3
(t) = a
31
(t)x
1
(t)+a
32
(t)x
2
(t)+¢¢¢+a
3n
(t)x
n
(t)+b
3
(t)
.
.
.
x
0
n
(t) = a
n1
(t)x
1
(t)+a
n2
(t)x
2
(t)+¢¢¢+a
nn
(t)x
n
(t)+f
1
(t)
Let x(t) be the column vector of unknown functions x
i
t; i =
1;2;¢¢¢ ;n, A(t) = (a
ij
(t); and b(t) be the column vector of known
functions b
i
t; i = 1;2;¢¢¢ ;n; we can write the ?rst order system of
equations as
x
0
(t)=A(t)x(t)+b(t) (1)
² When n =2, the linear ?rst order system of equations for two
unknown functions in matrix form is,
·
x
0
1
(t)
x
0
2
(t)
¸
=
·
a
11
(t) a
12
(t)
a
21
(t) a
22
(t)
¸·
x
1
(t)
x
2
(t)
¸
+
·
b
1
(t)
b
2
(t)
¸
² When n = 3, the linear ?rst order system of equations for
three unknown functions in matrix form is,
2
4
x
0
1
(t)
x
0
2
(t)
x
0
3
(t)
3
5
=
2
4
a
11
(t) a
12
(t) a
13
a
21
(t) a
22
(t) a
23
a
31
(t) a
32
(t) a
33
3
5
2
4
x
1
(t)
x
2
(t)
x
3
t
3
5
+
2
4
b
1
(t)
b
2
(t)
b
3
(t)
3
5
Asolution of equation (1) on the open interval I is a column vec-
tor function x(t) whose derivative (as a vector-values function) equals
A(t)x(t)+b(t): The following theorem gives existence and uniqueness
of solutions,
Theorem 1.1. Ifthevector-valuedfunctionsA(t)andb(t)arecon-
tinuous over an open interval I contains t
0
; then the initial value prob-
lem
½
x
0
(t)=A(t)x(t)+b(t)
x(t
0
)=x
0
has an unique vector-values solution x(t) that is de?ned on entire in-
terval I for any given initial value x
0
:
Whenb(t)´0; the linear ?rst order system of equations becomes
x
0
(t)=A(t)x(t);
which is called a homogeneous equation.
As in the case of one equation, we want to ?nd out the general
solutions for the linear ?rst order system of equations. To this end, we
?rst have the following results for the homogeneous equation,
Theorem1.2. PrincipleofSuperpositionLetx
1
(t); bx
2
(t);¢¢¢ ;x
n
(t)
be n solutions of the homogeneous linear equation
x
0
(t)=A(t)x(t)
on the open interval I: If c
1
; c
2
; ¢¢¢ ; c
n
are n constants, then the
linear combination
c
1
x
1
(t)+c
2
x
2
(t)+c
3
x
3
(t)+¢¢¢+c
n
x
n
(t)
is also a solution on I:
Example 1.1. Let
x
0
(t)=
·
1 0
0 ¡2
¸
x(t)
, x
1
(t)=
·
e
t
0
¸
and x
2
(t)=
·
0
e
¡2t
¸
are two solutions, as
bx
0
1
(t)=
·
(e
t
)
0
0
¸
=
·
e
t
0
¸
=
·
1 0
0 ¡2
¸·
e
t
0
¸
and
bx
0
2
(t)=
·
0
(e
¡2t
)
0
¸
=
·
0
¡2e
¡2t
¸
=
·
1 0
0 ¡2
¸·
0
e
¡2t
¸
By the Principle of Superposition, for any two constants c
1
and c
2
x(t)=c
1
x
1
(t)+c
2
x
2
(t)=c
1
·
e
t
0
¸
+c
2
·
0
e
¡2t
¸
=
·
c
1
e
t
c
2
e
¡2t
¸
is also solution. We shall see that it is actually the general solution.
The next theorem gives the general solution of linear system of
equations,
Theorem 1.3.
- Let x
1
(t); x
2
(t); ¢¢¢ ; bx
n
(t) be n linearly independent (as vectors)
solution of the homogeneous system
x
0
(t)=A(t)x(t);
then for any solutionx
c
(t) there exists n constants c
1
; c
2
; ¢¢¢ ; c
n
such
that
x
c
(t)=c
1
x
1
(t)+c
2
x
2
(t)+¢¢¢+c
n
x
n
(t):
We call x
c
(t) the general solution of the homogeneous system.
If x
p
(t) is a particular solution of the nonhomogeneous system,
x(t)=B(t)x(t)+b(t);
and x
c
(t) is the general solution to the associate homogeneous system,
x(t)=B(t)x(t)
then x(t)=x
c
(t)+x
p
(t) is the general solution.
Example 1.2. Let
x
0
(t)=
·
4 ¡3
6 ¡7
¸
x(t)+
·
¡4t
2
+5t
¡6t
2
+7t+1
¸
x(t)
,x
1
(t)=
·
3e
2t
2e
2t
¸
andx
2
(t)=
·
e
¡5t
3e
¡5t
¸
are two linearly independent
solutions. and x
p
(t) =
·
t
2
t
¸
is a particular solution. By Theorem
1.3,
x(t)=c
1
x
1
(t)+c
2
x
2
(t)+x
p
(t)=
·
3c
1
e
2t
+c
2
e
¡5t
+t
2
2c
1
e
2t
+3c
2
e
¡5t
+t
¸
(2)
is the general solution. Now suppose we want to ?nd a particular so-
lution that satis?es the initial condition x(0)=
·
2
¡1
¸
; then let t=0
in (2), we have
x(0)=
·
3c
1
+c
2
2c
1
+3c
2
¸
=
·
2
¡1
¸
;
which can be written in matrix form,
·
3 1
2 3
¸·
c
1
c
2
¸
=
·
2
¡1
¸
;
Solve this equation, we get
·
c
1
c
2
¸
=
·
1
¡1
¸
: So the particular
solution is x(t)=
·
3e
2t
¡e
¡5t
+t
2
2e
2t
¡3e
¡5t
+t
¸
:
From the above example, we can summarize the general steps in
?nd a solution to initial value problem,
½
x
0
(t)=A(t)x(t)+b(t)
x(t
0
)=x
0
Page 5


CHAPTER 1
System of First Order Di?erential Equations
Inthischapter,wewilldiscusssystemof?rstorderdi?erentialequa-
tions. Therearemanyapplicationsthatinvolving?ndseveralunknown
functions simultaneously . Those unknown functions are related by a
set of equations that involving the unknown functions and their ?rst
derivatives. For example, in Chapter Two, we studied the epidemic of
contagious diseases. Now if
² S(t)denotesnumberofpeoplethatissusceptibletothedisease
but not infected yet.
² I(t) denotes number of people actually infected.
² R(t) denotes the number of people have recovered.
If we assume
² The fraction of the susceptible who becomes infected per unit
time is proportional to the number infected, b is the propor-
tional number.
² A?xedfractionrS oftheinfectedpopulationrecoversperunit
time, 0·r· 1.
² A ?xed fraction of the recovers g become susceptible and in-
fected, 0·g· 1. proportional function.
The system of di?erential equations model this phenomena are
S
0
=¡bIS +gR
I
0
=bIS¡rI
R
0
=rI¡gR
The numbers of unknown function in a system of di?erential equa-
tions can be arbitrarily large, but we will concentrate ourselves on 2 to
3 unknown functions.
1. Principle of superposition
Let a
ij
(t); b
j
(t) i = 1;2;¢¢¢ ;n and j = 1;2;¢¢¢ ;n be known
function, and x
i
t; i =1;2;¢¢¢ ;nbeunknownfunctions, thelinear?rst
order system of di?erential equation for x
i
(t) is the following,
x
0
1
(t) = a
11
(t)x
1
(t)+a
12
(t)x
2
(t)+¢¢¢+a
1n
(t)x
n
(t)+b
1
(t)
x
0
2
(t) = a
21
(t)x
1
(t)+a
22
(t)x
2
(t)+¢¢¢+a
2n
(t)x
n
(t)+b
2
(t)
x
0
3
(t) = a
31
(t)x
1
(t)+a
32
(t)x
2
(t)+¢¢¢+a
3n
(t)x
n
(t)+b
3
(t)
.
.
.
x
0
n
(t) = a
n1
(t)x
1
(t)+a
n2
(t)x
2
(t)+¢¢¢+a
nn
(t)x
n
(t)+f
1
(t)
Let x(t) be the column vector of unknown functions x
i
t; i =
1;2;¢¢¢ ;n, A(t) = (a
ij
(t); and b(t) be the column vector of known
functions b
i
t; i = 1;2;¢¢¢ ;n; we can write the ?rst order system of
equations as
x
0
(t)=A(t)x(t)+b(t) (1)
² When n =2, the linear ?rst order system of equations for two
unknown functions in matrix form is,
·
x
0
1
(t)
x
0
2
(t)
¸
=
·
a
11
(t) a
12
(t)
a
21
(t) a
22
(t)
¸·
x
1
(t)
x
2
(t)
¸
+
·
b
1
(t)
b
2
(t)
¸
² When n = 3, the linear ?rst order system of equations for
three unknown functions in matrix form is,
2
4
x
0
1
(t)
x
0
2
(t)
x
0
3
(t)
3
5
=
2
4
a
11
(t) a
12
(t) a
13
a
21
(t) a
22
(t) a
23
a
31
(t) a
32
(t) a
33
3
5
2
4
x
1
(t)
x
2
(t)
x
3
t
3
5
+
2
4
b
1
(t)
b
2
(t)
b
3
(t)
3
5
Asolution of equation (1) on the open interval I is a column vec-
tor function x(t) whose derivative (as a vector-values function) equals
A(t)x(t)+b(t): The following theorem gives existence and uniqueness
of solutions,
Theorem 1.1. Ifthevector-valuedfunctionsA(t)andb(t)arecon-
tinuous over an open interval I contains t
0
; then the initial value prob-
lem
½
x
0
(t)=A(t)x(t)+b(t)
x(t
0
)=x
0
has an unique vector-values solution x(t) that is de?ned on entire in-
terval I for any given initial value x
0
:
Whenb(t)´0; the linear ?rst order system of equations becomes
x
0
(t)=A(t)x(t);
which is called a homogeneous equation.
As in the case of one equation, we want to ?nd out the general
solutions for the linear ?rst order system of equations. To this end, we
?rst have the following results for the homogeneous equation,
Theorem1.2. PrincipleofSuperpositionLetx
1
(t); bx
2
(t);¢¢¢ ;x
n
(t)
be n solutions of the homogeneous linear equation
x
0
(t)=A(t)x(t)
on the open interval I: If c
1
; c
2
; ¢¢¢ ; c
n
are n constants, then the
linear combination
c
1
x
1
(t)+c
2
x
2
(t)+c
3
x
3
(t)+¢¢¢+c
n
x
n
(t)
is also a solution on I:
Example 1.1. Let
x
0
(t)=
·
1 0
0 ¡2
¸
x(t)
, x
1
(t)=
·
e
t
0
¸
and x
2
(t)=
·
0
e
¡2t
¸
are two solutions, as
bx
0
1
(t)=
·
(e
t
)
0
0
¸
=
·
e
t
0
¸
=
·
1 0
0 ¡2
¸·
e
t
0
¸
and
bx
0
2
(t)=
·
0
(e
¡2t
)
0
¸
=
·
0
¡2e
¡2t
¸
=
·
1 0
0 ¡2
¸·
0
e
¡2t
¸
By the Principle of Superposition, for any two constants c
1
and c
2
x(t)=c
1
x
1
(t)+c
2
x
2
(t)=c
1
·
e
t
0
¸
+c
2
·
0
e
¡2t
¸
=
·
c
1
e
t
c
2
e
¡2t
¸
is also solution. We shall see that it is actually the general solution.
The next theorem gives the general solution of linear system of
equations,
Theorem 1.3.
- Let x
1
(t); x
2
(t); ¢¢¢ ; bx
n
(t) be n linearly independent (as vectors)
solution of the homogeneous system
x
0
(t)=A(t)x(t);
then for any solutionx
c
(t) there exists n constants c
1
; c
2
; ¢¢¢ ; c
n
such
that
x
c
(t)=c
1
x
1
(t)+c
2
x
2
(t)+¢¢¢+c
n
x
n
(t):
We call x
c
(t) the general solution of the homogeneous system.
If x
p
(t) is a particular solution of the nonhomogeneous system,
x(t)=B(t)x(t)+b(t);
and x
c
(t) is the general solution to the associate homogeneous system,
x(t)=B(t)x(t)
then x(t)=x
c
(t)+x
p
(t) is the general solution.
Example 1.2. Let
x
0
(t)=
·
4 ¡3
6 ¡7
¸
x(t)+
·
¡4t
2
+5t
¡6t
2
+7t+1
¸
x(t)
,x
1
(t)=
·
3e
2t
2e
2t
¸
andx
2
(t)=
·
e
¡5t
3e
¡5t
¸
are two linearly independent
solutions. and x
p
(t) =
·
t
2
t
¸
is a particular solution. By Theorem
1.3,
x(t)=c
1
x
1
(t)+c
2
x
2
(t)+x
p
(t)=
·
3c
1
e
2t
+c
2
e
¡5t
+t
2
2c
1
e
2t
+3c
2
e
¡5t
+t
¸
(2)
is the general solution. Now suppose we want to ?nd a particular so-
lution that satis?es the initial condition x(0)=
·
2
¡1
¸
; then let t=0
in (2), we have
x(0)=
·
3c
1
+c
2
2c
1
+3c
2
¸
=
·
2
¡1
¸
;
which can be written in matrix form,
·
3 1
2 3
¸·
c
1
c
2
¸
=
·
2
¡1
¸
;
Solve this equation, we get
·
c
1
c
2
¸
=
·
1
¡1
¸
: So the particular
solution is x(t)=
·
3e
2t
¡e
¡5t
+t
2
2e
2t
¡3e
¡5t
+t
¸
:
From the above example, we can summarize the general steps in
?nd a solution to initial value problem,
½
x
0
(t)=A(t)x(t)+b(t)
x(t
0
)=x
0
² StepOne: Findthegeneralsolutionx
c
=c
1
x
1
(t)+c
2
x
2
(t)+
¢¢¢+c
n
x
n
(t); where x
1
(t); x
2
(t); ¢¢¢ ; x
n
(t) are a set of lin-
earlyindependentsolutions,totheassociatehomogeneoussys-
tem,x
0
(t)=A(t)x(t):
² Step Two: Find a particular solution x
p
(t)to the nonhomo-
geneous system, x
0
(t)=A(t)x(t)+b(t):
² Step Three: Set x(t) = x
c
(t)+x
p
(t) and use the equation
x(t
0
)=x
0
; to determine c
1
; c
2
; ¢¢¢ ; c
n
:
2. Homogeneous System
We will use a powerful method called eigenvalue method to solve
the homogeneous system
x
0
(t)=Ax(t)
whereA is a matrix with constant entry. We will present this method
for A is either a 2£2 or 3£3 cases. The method can be used for A
is an n£n matrix. The idea is to ?nd solutions of form
x(t)=ve
¸t
; (3)
a straight line that passing origin in the direction v: Now taking deriv-
ative onx(t); we have
x
0
(t)=¸ve
¸t
(4)
put (3) and (2.2) into the homogeneous equation, we get
x
0
(t)=¸ve
¸t
=Ave
¸t
So
Av =¸v;
whichindicatesthat¸mustbeaneigenvalueofAandv isanassociate
eigenvector.
2.1. A is a 2£2 matrix. Suppose
A =
·
a
11
a
12
a
21
a
22
¸
Then the characteristic polynomial p(¸) ofA is
p(¸)=jA¡¸Ij=(a
11
¡¸)¤(a
22
¡¸)¡a
12
a
21
=¸
2
¡(a
11
+a
22
)+(a
11
a
22
¡a
12
a
22
:
Sop(¸)isaquadraticpolynomialof ¸:FromAlgebra,weknowthat
p(¸) = 0 has either 2 distinct real solutions, or a double solution, or
2 conjugate complex solutions. The following theorem summarize the
solution to the homogeneous system,