LECTURE 28: NONHOMOGENEOUS SYSTEM AND VECTOR FIELDS FOR SYSTEM
MINGFENG ZHAO
November 20, 2015
Theorem 1. Let ~xc (t) be the general solution to the homogeneous system ~x0 = A~x, and ~xp (t) be a particular solution
to ~x0 = A~x + f~(t), then the general solution to ~x0 = A~x + f~(t) is:
~x(t) = ~xc (t) + ~xp (t).
Variation of Parameters

Let A be a 2 × 2 matrix, and ~y (t) = 
y1 (t)


 and ~z(t) = 
y2 (t)

~x0 = A(t)~x, let X(t) = [~y (t) ~z(t)] = 
y1 (t) z1 (t)
z1 (t)

 be two linearly independent solutions to
z2 (t)

, then X(t) is a fundamental matrix, and X 0 (t) = A(t)X(t). Hence
y2 (t) z2 (t)
the general solution to ~x0 = A(t)~x is:

~x(t) = C1 ~y (t) + C2 ~z(t) = [~y (t) ~z(t)] 
C1
C2


=
y1 (t) z1 (t)

C1

y2 (t) z2 (t)

~
 = X(t)C.
C2
Let’s consider
~x0 = A(t)~x + f~(t).
Let ~xp (t) = X(t)~u(t) for some ~u(t) be a particular solution to ~x0 = A(t)~x + f~(t), then
~x0p = X 0 (t)~u(t) + X(t)~u0 (t) = A(t)~xp + f~(t) = A(t)X(t)~u(t) + f~(t).
Since X 0 (t) = A(t)X(t), then
A(t)X(t)~u(t) + X(t)~u0 (t) = A(t)X(t)~u(t) + f~(t).
That is, X(t)~u0 (t) = f~(t), which implies that
~u0 (t) = X(t)−1 f~(t).
1
2
MINGFENG ZHAO
Then
Z
~u(t) =
X(t)−1 f~(t) dt.
That is,
Z
~xp (t) = X(t)

Remark 1. Let A = 
a
b
c
d

, then
A−1

Example 1. Let A = 
X(t)−1 f~(t) dt.
−1
0
−2
1



d
1  d −b 
1

=
=
det A −c a
ad − bc −c
−b

.
a


. Find a particular solution of ~x0 = A~x + f~(t), where f~(t) = 
et

.
t

First, let’s find the eigenvalues of A, that is, det (A − λI2 ) = det 
−1 − λ
0
−2
1−λ

 = (−λ − 1)(1 − λ) = 0, then
and λ2 = −1.
λ1 = 1,
For λ1 = 1, let’s solve A~x = ~x, that is,



Then 
x1
x2


 = x2 
0


, that is, 
1
0
−2
0
−2
0

x1



=
x2
0

.
0

 is an eigenvalue corresponding to λ = 1, which implies that
1

0



 et = 
1
0
e
t

 is a solution to ~x0 = A~x.
For λ2 = −1, let’s solve A~x = −~x, that is,


0
0
−2
2

x1

x2


=
0
0

.
LECTURE 28: NONHOMOGENEOUS SYSTEM AND VECTOR FIELDS FOR SYSTEM

Then 
x1
x2


 = x1 
1


, tha is, 
1
1

 is an eigenvalue corresponding to λ = −2, which implies that
1

1



 e−t = 
1
e−t
e
−t

 is a solution to ~x0 = A~x.
Then we can take the fundamental matrix X(t) as:

X(t) = 
0
e−t
t
−t
e
e

.
Then det X(t) = −1, and

X(t)−1 = − 
e−t
−e−t
−et
0


=
−e−t
e−t
et
0

.
Let ~xp (t) = X(t)~u(t) for some ~u(t) be a particular solution to ~x0 = A(t)~x + f~(t), then
~x0p = X 0 (t)~u(t) + X(t)~u0 (t) = A(t)~xp + f~(t) = A(t)X(t)~u(t) + f~(t).
Since X 0 (t) = AX(t), then X(t)~u0 (t) = f~(t), that is, ~u0 (t) + X(t)−1 f~(t). So we get
Z
~u(t) =
X(t)−1 f~(t) dt

Z
=


Z
=
−e−t
e−t
et
0
−1 + te−t
e
= 
et




2t
 dt
t

 dt
−t − te−t − e−t
1 2t
2e


.
Then
~x(t)
=
=
=
X(t)~u(t)



0 e−t
−t − te−t − e−t



1 2t
et e−t
e
2


1 t
e
2

.
t
−te − t − 1 + 12 et
3
4
MINGFENG ZHAO
So a particular solution to ~x0 = A~x + f~(t) is:

~xp (t) = 

Example 2. Let A = 
1
−1
2
−1

1 t
2e
−tet − t − 1 + 12 et
.


. Find a particular solution of ~x0 = A~x + f~(t), where f~(t) = 
0

.
1

First, let’s find the eigenvalues of A, that is, det (A−λI2 ) = det 
1−λ
−1
2
−1 − λ

 = (1−λ)(−1−λ)+2 = λ2 +1 = 0,
then
λ1 = i,
and λ2 = −i.
For λ1 = i, let solve A~x = i~x, that is,

1−i
−1
2
−1 − i


Then 
x1


 = x1 
x2

eit 
1
1−i
1
1−i


, that is, 



= 
= 


=
x2
0

.
0



cos(t) + i sin(t)
(cos(t) + i sin(t))(1 − i)


cos(t) + i sin(t)
= 

 is an eigenvalue corresponding to λ = i, which implies that
1−i
eit (1 − i)

x1

1
eit



cos(t) + sin(t) + i sin(t) − i cos(t)




cos(t)
sin(t)
 + i
 is a solution to ~x0 = A~x.
= 
sin(t) + sin(t)
sin(t) − cos(t)
So the general solution to ~x0 = A~x is:

~xc (t) = C1 
cos(t)
sin(t) + sin(t)


 + C2 
sin(t)
sin(t) − cos(t)

.
LECTURE 28: NONHOMOGENEOUS SYSTEM AND VECTOR FIELDS FOR SYSTEM
Then we can take the fundamental matrix X(t) as:

cos(t)
X(t) = 
cos(t) + sin(t)
sin(t)
sin(t) − cos(t)
5

.
Then det X(t) = −1, and

X(t)−1 = − 
sin(t) − cos(t)
− sin(t)
− cos(t) − sin(t)
cos(t)


=
cos(t) − sin(t)
sin(t)
cos(t) + sin(t) − cos(t)

.
Then
Z
~x(t)
=
X(t)

=

X(t)−1 f~(t) dt
cos(t)
=


=


=


=

sin(t) − cos(t)


Z

cos(t) − sin(t)

sin(t)
cos(t) + sin(t) − cos(t)
 

Z
cos(t)
sin(t)
sin(t)
 
 dt
cos(t) + sin(t) sin(t) − cos(t)
− cos(t)


cos(t)
sin(t)
− cos(t)


cos(t) + sin(t) sin(t) − cos(t)
− sin(t)

− cos2 (t) − sin2 (t)

− cos2 (t) − sin(t) cos(t) − sin2 (t) + sin(t) cos(t)

−1

−1
cos(t) + sin(t)

sin(t)
So a particular solution to ~x0 = A~x + f~(t) is:

~xp (t) = 
−1
−1

.
Two dimensional systems and theirvector fields
To draw the vector field of a general system
 x0 = f1 (t, x1 , x2 )
1
:
 x0 = f (t, x , x )
2
1
2
2
I. Plot the tx1 x2 -space.
II. Select points as many as possible in the plane, say P1 , P2 , · · · , Pn .

0

1

 dt
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MINGFENG ZHAO
III. At each point Pi , draw a short arrow with direction (1, f1 (Pi ), f2 (Pi )).
In this course, we only study the two dimensional autonomous system:

 x0 = f1 (x1 , x2 )
1
 x0 = f (x , x )
2
2
1
2

 x0 = f1 (x1 , x2 )
1
The vector field of
is the projection on the x1 x2 -plane of its three dimensional vector field in the
 x0 = f (x , x )
2
1
2
2

 x0 = f1 (x1 , x2 )
1
tx1 x2 -space. To draw the vector field of
:
 x0 = f (x , x )
2
1
2
2
I. Plot the x1 x2 -plane.
II. Select points as many as possible in the plane, say P1 , P2 , · · · , Pn .
III. At each point Pi , draw a short arrow with direction (f1 (Pi ), f2 (Pi )).
Problems you can do:
Lebl’s Book [2]: Do all Exercises on Page 146 and Page 147 using variation of parameters.
Braun’s Book [1]: All exercises on Page 367 and Page 368. Read all materials in Section 3.12.
References
[1] Martin Braun. Differential Equations and Their Applications: An Introduction to Applied Mathematics. Springer, 1992.
[2] Jiri Lebl. Notes on Diffy Qs: Differential Equations for Engineers. Createspace, 2014.
Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C.
Canada V6T 1Z2
E-mail address: mingfeng@math.ubc.ca