LECTURE 23: SYSTEMS OF ODES
MINGFENG ZHAO
October 29, 2014
Introduction to systems of ODEs
In this course, we only study the systems of the first order differential equations which has the form:

 x0 (t) = f1 (t, x1 , x2 )
1
 x0 (x) = f (t, x , x ).
2
2
1
2
A second order differential equation can be viewed as a system of the first order differential equations, e.g., x00 =
f (t, x, x0 ). Let x1 (t) = x(t) and x2 (t) = x0 (t), then

 x0 (t) = x0 (t) = x2
1
 x0 (t) = x00 (t) = f (t, x, x0 ) = f (t, x , x ).
1
2
2
Remark 1. In general, an n-th order differential equation can be transformed into a first order system with n unknown
functions.
In the Chapter 3, we only study the linear system in the following form:

 x0 = ax1 + bx2 + f1 (t)
1
 x0 = cx + dx + f (t)
1
2
2
2

Let ~x(t) = 
x1 (t)
x2 (t)

,

A=
a
b
c
d

,

and f~(t) = 
f1 (t)

, then we have
f2 (t)
~x0 = A~x + f~(t) .
The matrix A is called the coefficient matrix of the system, and f~(t) is called the non-homogeneous term of the
system. If f~(t) ≡ 0, we say ~x0 = A~x is homogeneous.
1
2
MINGFENG ZHAO

Let A = 
a
b
c
d

, then

det A = ad − bc,
A−1 =
and
−b
d
1

ad − bc −c

 , if ad − bc 6= 0.
a
Theorem 1. Let ~x0 = A~x + f~(t) be a linear system of ODEs, and ~xp (t) be a particular solution to ~x0 = A~x + f~(t), then
the general solution to ~x0 = A~x + f~(t) has the form of:
~x(t) = ~xc (t) + ~xp (t),
where ~xc (t) is any solution to the homogeneous system ~x0 = A~x.
Homogeneous system
Question 1. How can we solve ~x0 = A~x?
Theorem 2 (Superposition). Let ~y and ~z be two solutions to ~x0 = A~x, then w
~ = C1 ~y +C2 ~z is also a solution to ~x0 = A~x.
Definition 1. Let ~x1 (t), ~xn (t) be n many vector functions, we say that ~x1 (t), ~xn (t) are linearly dependent if there exist
some constants c1 , · · · , cn such that not all ci ’s are zero, and
c1 ~x1 (t) + · · · + c2 ~x2 (t) = 0.
If ~x1 (t), · · · , ~xn (t) are not linearly dependent, then we say they are linearly independent.

Example 1. Let ~x1 (t) = 
t2


, ~x2 (t) = 
t

0

, and ~x3 (t) = 
−t2
1+t

, then it’s easy to see that ~x1 (t) =
1
~x2 (t) − ~x3 (t), that is, ~x1 (t), ~x2 (t), ~x3 (t) are linearly dependent .
General Approach: Let C1 , C2 and C3 be such that
C1 ~x1 (t) + C2 ~x2 (t) + C3 ~x3 (t) = 0.
That is,

C1 
t2
t


 + C2 
0
1+t


 + C3 
−t2
1


=
0
0

.
LECTURE 23: SYSTEMS OF ODES
3
So we get


C1 t2 − C3 t2

=

C1 t + C2 (1 + t) + C3
0

.
0
Then
C1 t2 − C3 t2 = 0,
and C1 t + C2 (1 + t) + C3 = 0.
So we get
C1 = C3 ,
C1 + C2 = 0,
and C2 + C3 = 0.
Then
C1 = C3 = −C2 .
If we take C1 = 1, then C2 = −1 and C3 = 1, that is, ~x1 (t) − ~x2 (t) + ~x3 (t) = 0, which implies that
~x1 (t), ~x2 (t), ~x3 (t) are linearly dependent .

Example 2. Let ~x1 (t) = 
1
1


 et , and ~x2 (t) = 
1
−1

 et , it’s easy to see that we can not find a constant C such that
~x1 (t) = C~x2 (t) or ~x2 (t) = C~x1 (t), that is, ~x1 (t), ~x2 (t) are linearly independent .
Eigenvalue method for the general homogeneous system
Theorem 3. Let ~y (t) and ~z(t) be two linearly independent solutions to ~x0 = A~x, then the general solution to ~x0 = A~x is
~x(t) = C1 ~y (t) + C2 ~z(t).
Definition 2. Let A be a 2 × 2 matrix, suppose that there is a scalar λ and a nonzero vector ~v such that
A~v = λ~v ,
in this case, λ is called an eigenvalue of A, and ~v is called an eigenvector corresponding to λ.

Theorem 4. Let A be a 2 × 2 matrix, λ is an eigenvalue of A if and only if det (λI2 − A) = 0, where I2 = 
1
0
0
1

.
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MINGFENG ZHAO

Remark 2. Notice that if A = 
a
b
c
d

, then

det (λI2 − A)
=
det 
λ−a
−c
−b
λ−d


=
(λ − a)(λ − d) − c
=
λ2 − (a + d)λ + ad − bc.
Theorem 5. Let λ be an eigenvalue of A and ~v be an eigenvector corresponding to λ, then ~x(t) = eλt~v is a solution to
~x0 = A~x.
Proof. In fact, we have
~x0 (t) = λeλt~v = eλt A~v = A(eλt~v ) = A~x(t).
Remark 3. For eigenvalues λ of a 2 × 2 marix A, we have three cases:
I. A has two different real roots λ1 and λ2 , in this case, eigenvectors ~v 1 and ~v 2 are real.
II. A has the same real root λ, in this case, eigenvector ~v is real.
III. A hs two different complex roots λ1 and λ2 , in this case, eigenvectors ~v 1 and ~v 2 are complex.
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