MA282 – Spring 2013
10.1 Theory of Linear Systems
Introduction
In this chapter we confine our study to systems of first-order DEs that are special cases of
systems that have the normal form.
Definitions
A system of 𝑛 first-order equations is called a first-order system.
𝑑𝑥1
= 𝑔1 (𝑡, 𝑥1 , 𝑥2 , ⋯ , 𝑥𝑛 )
𝑑𝑡
𝑑𝑥2
= 𝑔2 (𝑡, 𝑥1 , 𝑥2 , ⋯ , 𝑥𝑛 )
𝑑𝑡
⋮
𝑑𝑥𝑛
= 𝑔𝑛 (𝑡, 𝑥1 , 𝑥2 , ⋯ , 𝑥𝑛 )
𝑑𝑡
When each of the functions 𝑔1 , 𝑔2 , … , 𝑔𝑛 is linear in the dependent variables 𝑥1 , 𝑥2 , … , 𝑥𝑛 ,
we get the normal form of a first-order system of linear equations:
𝑑𝑥1
= 𝑎11 (𝑡)𝑥1 + 𝑎12 (𝑡)𝑥2 + ⋯ + 𝑎1𝑛 (𝑡)𝑥𝑛 + 𝑓1 (𝑡)
𝑑𝑡
𝑑𝑥2
= 𝑎21 (𝑡)𝑥1 + 𝑎22 (𝑡)𝑥2 + ⋯ + 𝑎2𝑛 (𝑡)𝑥𝑛 + 𝑓2 (𝑡)
𝑑𝑡
⋮
𝑑𝑥𝑛
= 𝑎𝑛1 (𝑡)𝑥1 + 𝑎𝑛2 (𝑡)𝑥2 + ⋯ + 𝑎𝑛𝑛 (𝑡)𝑥𝑛 + 𝑓𝑛 (𝑡)
𝑑𝑡
We refer to a system of the form above simply as a linear system. We assume that the
coefficients 𝑎𝑖𝑗 (𝑡) as well as the functions 𝑓𝑖 (𝑡) are continuous on a common interval 𝐼.
When 𝑓𝑖 (𝑡) = 0, 𝑖 = 1, 2, ⋯ , 𝑛, the linear system is said to be homogeneous; otherwise it is
nonhomogeneous.
Matrix Form of a Linear System
If 𝑿, 𝑨(𝑡), and 𝑭(𝑡) denote the respective matrices
𝑎11 (𝑡) ⋯
𝑥1 (𝑡)
⋱
𝑋 = ( ⋮ ) , 𝐴(𝑡) = ( ⋮
𝑎𝑛1 (𝑡) ⋯
𝑥𝑛 (𝑡)
𝑎1𝑛 (𝑡)
𝑓1 (𝑡)
⋮ ) , 𝐹(𝑡) = ( ⋮ )
𝑎𝑛𝑛 (𝑡)
𝑓𝑛 (𝑡)
then the system of linear first-order differential equations can be written as
𝑥1
𝑎11 (𝑡) ⋯
𝑑
⋮
⋱
( )=( ⋮
𝑑𝑡 𝑥
𝑎𝑛1 (𝑡) ⋯
𝑛
𝑎1𝑛 (𝑡) 𝑥1
𝑓1 (𝑡)
⋮
⋮ )( ) + ( ⋮ )
𝑎𝑛𝑛 (𝑡) 𝑥𝑛
𝑓𝑛 (𝑡)
𝑿′ = 𝑨𝑿 + 𝑭
or simply
(𝟏)
If the system is homogeneous, its matrix form is then
𝑿′ = 𝑨𝑿
Definition
A solution vector on an interval 𝐼 is any column matrix
𝑥1 (𝑡)
𝑋=( ⋮ )
𝑥𝑛 (𝑡)
whose entries are differentiable functions satisfying the system on the interval.
Example Write the system in Matrix form:
dx
= 3x + 4y
dt
dy
= 5x − 7y
dt
Example Verify that on the interval (−∞, ∞),
3
1
) 𝑒 −2𝑡 and 𝑋2 = ( ) 𝑒 6𝑡
5
−1
𝑋1 = (
are solutions of
𝑋′ = (
1 3
)𝑋
5 3
(𝟐)
Definition Let 𝑡0 denote a point on an interval 𝐼 and
𝛾1
𝑥1 (𝑡0 )
𝑋(𝑡0 ) = ( ⋮ ) and 𝑋0 = ( ⋮ )
𝛾𝑛
𝑥𝑛 (𝑡0 )
where the 𝛾𝑖 , 𝑖 = 1, 2, ⋯ , 𝑛 are given constants. Then the problem
Solve: 𝑋 ′ = 𝐴(𝑡)𝑋 + 𝐹(𝑡)
Subject to: 𝑋(𝑡0 ) = 𝑋0
is an initial-value problem on the interval.
Theorem Existence and Uniqueness of Solution
Let the entries of the matrices 𝐴(𝑡) and 𝐹(𝑡) be functions continuous on a common interval
𝐼
that contains the point 𝑡0 . Then there exists a unique solution of the initial-value problem on
the interval.
Theorem Superposition Principle
Let 𝑋1 , 𝑋2 , ⋯ , 𝑋𝑘 be a set of solution vectors of the homogeneous system (2) on an interval 𝐼.
Then the linear combination
𝑋 = 𝑐1 𝑋1 + 𝑐2 𝑋2 + ⋯ + 𝑐𝑘 𝑋𝑘 ,
is also a solution on the interval where the 𝑐𝑖 , 𝑖 = 1,2, ⋯ , 𝑘 are arbitrary constants.
We are primarily interested in linearly independent solutions of the homogeneous system (2).
Definition
Let 𝑋1 , 𝑋2 , ⋯ , 𝑋𝑘 be a set of solution vectors of the homogeneous system (2) on an interval 𝐼.
We say that the set is linearly dependent on the interval if there exist constants 𝑐1 , 𝑐2 , ⋯ , 𝑐𝑘
not all zero, such that
𝑐1 𝑋1 + 𝑐2 𝑋2 + ⋯ + 𝑐𝑘 𝑋𝑘 = 0
for every 𝑡 in the interval. If the set of vectors is not linearly dependent on the interval, it is
said to be linearly independent.
As in our earlier consideration of the theory of a single ordinary differential equation in §3.1,
we can introduce the concept of the Wronskian determinant as a test for linear independence.
We state the following theorem without proof.
Theorem Criterion for Linearly Independent Solutions
𝑥11
𝑥12
𝑥1𝑛
Let
𝑋1 = ( ⋮ ) ,
𝑋2 = ( ⋮ ) , ⋯ , 𝑋𝑛 = ( ⋮ )
𝑥𝑛1
𝑥𝑛2
𝑥𝑛𝑛
be 𝑛 solutions vectors of the homogeneous system (2) on an interval 𝐼. Then the set of
solution vectors is linearly independent if and only if the Wronskian
𝑥11 ⋯ 𝑥1𝑛
⋱
⋮ |≠0
𝑊(𝑋1, 𝑋2 , ⋯ , 𝑋𝑛 ) = | ⋮
𝑥𝑛1 ⋯ 𝑥𝑛𝑛
for every 𝑡 in the interval.
Notice that, unlike our definition of the Wronskian in Section 3.1, here the definition of the
determinant does not involve differentiation.
Example We saw that
3
1
) 𝑒 −2𝑡 and 𝑋2 = ( ) 𝑒 6𝑡
5
−1
𝑋1 = (
are solutions of the system
𝑋′ = (
1 3
)𝑋
5 3
Easily we can check that 𝑋1 and 𝑋2 are linearly independent on (−∞, ∞) since neither
vector is a constant multiple of the other. In addition,
Definition Any set 𝑋1 , 𝑋2 , … , 𝑋𝑛 of 𝑛 linearly independent solution vectors of the
homogeneous system (2) on an interval 𝐼 is said to be a fundamental set of solutions on the
interval.
Theorem General Solution - Homogeneous Systems
Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be a fundamental set of solutions of the homogeneous system (2) on an
interval 𝐼. Then the general solution of the system on the interval is
𝑋 = 𝑐1 𝑋1 + 𝑐2 𝑋2 + ⋯ + 𝑐𝑛 𝑋𝑛 ,
where the 𝑐𝑖 , 𝑖 = 1, 2, … , 𝑛 are arbitrary constants.
Example We know that
3
1
) 𝑒 −2𝑡 and 𝑋2 = ( ) 𝑒 6𝑡
5
−1
𝑋1 = (
are linearly independent solutions of the system
𝑋′ = (
1 3
)𝑋
5 3
The general solution of the system on the interval is then
3
1
) 𝑒 −2𝑡 + 𝑐2 ( ) 𝑒 6𝑡
5
−1
𝑋 = 𝑐1 𝑋1 + 𝑐2 𝑋2 = 𝑐1 (
For nonhomogeneous systems, a particular solution 𝑋𝑝 on an interval 𝐼 is any vector, free of
arbitrary parameters, whose entries are functions that satisfy system (1).
Theorem General Solution-Nonhomogeneous Systems
Let 𝑋𝑝 be a given solution of the nonhomogeneous system (1) on an interval 𝐼, and let
𝑋𝑐 = 𝑐1 𝑋1 + 𝑐2 𝑋2 + ⋯ + 𝑐𝑛 𝑋𝑛
denote the general solution on the same interval of the associated homogeneous system (2).
Then the general solution of the nonhomogeneous system on the interval is
𝑋 = 𝑋𝑐 + 𝑋𝑝
The general solution 𝑋𝑐 of the associated homogeneous system is called the complementary
function of the nonhomogeneous system.
Example
Verify that the vector 𝑋𝑝 = (
3𝑡 − 4
) is a particular solution of the nonhomogeneous system
−5𝑡 + 6
1 3
12t − 11
X′ = (
)X+ (
)
5 3
−3
on the interval (−∞, ∞). The complementary function of the above nohomogeneous system
1 3
on the same interval, or the general solution of X ′ = (
) X was seen to be
5 3
3
1
𝑋𝑐 = 𝑐1 ( ) 𝑒 −2𝑡 + 𝑐2 ( ) 𝑒 6𝑡
5
−1
Hence by Theorem
3𝑡 − 4
3
1
) 𝑒 −2𝑡 + 𝑐2 ( ) 𝑒 6𝑡 + (
)
−5𝑡 + 6
5
−1
𝑋 = 𝑋𝑐 + 𝑋𝑝 = 𝑐1 (
is the general solution of the nohomogeneous system on the interval (−∞, ∞).