Differential Equations
Chapter 06:
Systems of
First Order
Linear
Equations
Brannan
Copyright © 2010 by John Wiley & Sons, Inc.
All rights reserved.
Chapter 6 Systems of First
Order Linear Equations
we build on elementary theory and solution
techniques for first order linear systems
 Science and engineering applications that
possess even a modest degree of complexity
often lead to systems of differential equations
of dimension
 n > 2.A large portion of Chapter 6 generalizes
the eigenvalue method presented in Chapter
3 to constant coefficient linear systems of
dimension n > 2.

Chapter 6 - Systems of First
Order Linear Equations







6.1 Definitions and Examples
6.2 Basic Theory of First Order Linear
Systems
6.3 Homogeneous Linear Systems with
Constant Coefficients
6.4 Nondefective Matrices with Complex
Eigenvalues
6.5 Fundamental Matrices and the
Exponential of a Matrix
6.6 Nonhomogeneous Linear Systems
6.7 Defective Matrices
6.1 Definitions and Examples
Matrix-Valued Functions.
The principal mathematical objects involved in the study of linear
systems of differential equations are matrix-valued functions, or
simply matrix functions. These
objects are vectors or matrices
whose elements are functions of t.
We write
The matrix P = P(t) is said to be continuous at t = t0 or on an interval I
= (α, β) if each element of P is a continuous function at the given
point or on the given interval. Similarly, P(t) is said to be differentiable
if each of its elements is differentiable, and its derivative dP/dt is
defined by
Similarly, the integral of a
matrix function is defined as
First Order Linear Systems:
General Framework
The general form of a first order linear system of
dimension n is
x'1= p11(t)x1 + p12(t)x2 + ・・・ + p1n(t)xn + g1(t),
x'2= p21(t)x1 + p22(t)x2 + ・・・ + p2n(t)xn + g2(t),
...
x'n= pn1(t)x1 + pn2 (t)x2 + ・・・ + pnn(t)xn + gn(t)
or using matrix notation,
x' = P(t)x + g(t),
 P(t) is matrix of coefficients. g(t) the
nonhomogeneous term of the system. If g(t) = 0
for all t ∈ I , then the system is said to be
homogeneous; otherwise the system is said to be
nonhomogeneous. The function g(t) is often
referred to as the input, or forcing function.

First Order Linear Systems:
General Framework (Ctd.)




The system is said to have a solution on the
interval I if there exists a vector x = φ(t) with n
components that is differentiable at all points in the
interval I and satisfies Eq. at all points in this
interval.
In addition to the system of differential equations, if
there is an initial condition of the form x(t0) = x0,
where t0 is a specified value of t in I and x0 is a
given constant vector with n components.
The system and the initial condition together form
an initial value problem.
The components of x are again referred to as state
variables and the vector x = φ(t) is referred to as
the state of the system at time t.
Linear nth Order Equations.

An nth order linear differential equation in
standard form is given by
to obtain a unique solution, we need n initial
conditions
 To transform Eq. into a system of n first order
equations, we introduce the variables x1, x2, . . .
, xn defined by x1 = y, x2 = y', x3 = y'' , ..., xn =
y(n−1).
 In Matrix

Applications Modeled by First
Order Linear Systems

1.
2.
3.
4.
We discuss several examples of
applications that illustrate how
higher dimensional linear systems
can arise.
Coupled Mass-Spring Systems.
Compartment Models.
Linear Control Systems.
The State Variable Approach to
Circuit Analysis.
6.2 Basic Theory of First Order
Linear Systems
THEOREM 6.2.1 (Existence and
Uniqueness for First Order Linear
Systems).
If P(t) and g(t) are continuous on an open
interval I = (α, β), then there exists a unique
solution x = φ(t) to the initial value problem
x' = P(t)x + g(t), x(t0) = x0, where t0 is any
point in I, and x0 is any constant vector with
n components. Moreover the solution exists
throughout the interval I.

THEOREM 6.2.2 (Principle of
Superposition).
If x1, x2, . . . , xk are solutions of the
homogeneous linear system
x' = P(t)x
on the interval I = (α, β), then the linear
combination
c1x1 + c2x2 + ・・・ + ckxk
is also a solution of the system on I.
DEFINITION 6.2.3
The n vector functions x1, x2, . . . , xn are said
to be linearly independent on an interval
I if the only constants c1, c2, . . . , cn such
that c1x1(t) + ・・・ + cnxn(t) = 0 for all t ∈ I
are c1=c2=. . . =cn = 0.
If there exist constants c1, c2, . . . , cn , not all
zero, such that c1x1(t) + ・・・ + cnxn(t) = 0
is true for all t ∈ I , the vector functions are
said to be linearly dependent on I.
Examples

1

2
DEFINITION 6.2.4

Let x1, . . . , xn be n solutions of x' = P(t)x
and let X(t) be the n×n matrix whose j th
column is xj (t), j = 1, . . . , n,

The Wronskian W = W[x1, . . . , xn] of the n
solutions x1, . . . , xn is defined by
W[x1, . . . , xn](t) = det X(t).
THEOREM 6.2.5
Let x1, . . . , xn be solutions of x' = P(t)x
on an interval I = (α, β) in which P(t) is
continuous.
(i) If x1, . . . , xn are linearly independent
on I, then
W[x1, . . . , xn](t) ≠ 0 at every point in I,
(ii) If x1, . . . , xn are linearly dependent
on I, then
W[x1, . . . , xn](t) = 0 at every point in I.
THEOREM 6.2.6
Example
Theorem 6.2.7
Linear nth Order Equations.
COROLLARY 6.2.8
If the functions p1(t), p2(t), . . . , pn(t), and g(t) are
continuous on the open intervalI = (α, β), then
there exists exactly one solution y = φ(t) of the
differential equation
that also satisfies the initial conditions
This solution exists throughout the interval I.
COROLLARY 6.2.9
A fundamental set of solutions
and the general solution
The terminology for solutions of the nth order
scalar equation is identical to that used for
solutions of x' = P(t)x. A set of solutions y1, . . . , yn
such that W[ y1, . . . , yn](t0) ≠ 0 for some t0∈ (α, β)
is called a fundamental set of solutions for Eq.
(28).
 The n-parameter family represented by the linear
combination
y = c1y1(t) + ・・ ・+cnyn(t),
where c1, . . . , cn are arbitrary constants, is called the
general solution of Eq. (28).
 Corollary 6.2.9 guarantees that each solution of Eq.
(28) corresponds to some member of this nparameter family of solutions.

Example
Q: Show that y1(x)=x, y2(x)=x−1, and y3(x)=x2
constitute a fundamental set of solutions for
x3 y'' + x2 y'' − 2xy' + 2y = 0 on I = (0,∞).
A:
1. Substituting each of y1, y2, and y3 into Eq.
show that they are solutions on the given
interval.
2. Verify that the three functions are linearly
independent on I, by computing the
Wronskian
3. Apply Corollary 6.2.9.
6.3 Homogeneous Linear Systems
with Constant Coefficients
In this section, we extend the eigenvalue
method to the system x' = Ax, where A is a
real constant n × n matrix. As in Chapter 3,
we assume solutions of the form x = eλtv,
where the scalar λ and the constant n × 1
vector v are to be determined.
 Given a square matrix A, recall that the
problem of
(i) finding values of λ for which Eq. (A − λIn)v
= 0 has nontrivial solution vectors v, and
(ii) finding the corresponding nontrivial
solutions, is known as the eigenvalue
problem for A.

Three cases
1. A has a complete set of n linearly
independent eigenvectors and all of the
eigenvalues of A are real,
 2. A has a complete set of n linearly
independent eigenvectors and one or more
complex conjugate pairs of eigenvalues,
 3. A is defective, that is, there are one or
more eigenvalues of A for which the
geometric multiplicity is less than the
algebraic multiplicity (see Appendix A.4).

The Matrix A Is Nondefective With
Real Eigenvalues

THEOREM 6.3.1

Example
Real and Distinct Eigenvalues.

COROLLARY 6.3.2

EXAMPLE
Symmetric Matrices.
Properties of the eigenvalues and
eigenvectors of a real symmetric matrix A
(see Appendix A.4):
1. All the eigenvalues λ1, . . . , λn of A are real.
2. A has a complete set of n real and linearly
independent eigenvectors v1, . . . , vn.
 Furthermore eigenvectors corresponding to
different eigenvalues are orthogonal to one
another and all eigenvectors belonging to
the same eigenvalue can be chosen to be
orthogonal to one another.

Example

Diffusion on a One-Dimensional Lattice
with Reflecting Boundaries.

the system describing diffusion on a lattice
consisting of n = 3 points is expressed in
matrix notation as
6.4 Nondefective Matrices with
Complex Eigenvalues
In this section, we again consider a system of
n linear homogeneous equations with constant
coefficients, x' = Ax, where the coefficient
matrix A is real-valued, nondefective, and has
one or more complex eigenvalues.
 If λ = μ + iν and (λ, v) is an eigenpair of A and
v = a + ib, where a and b are real constant n×1
vectors, then the vectors
x1(t) = Re u(t) = eμt (a cos νt − b sin νt)
x2(t) = Im u(t) = eμt (a sin νt + b cos νt)
are real-valued solutions of Eq. and are linearly
independent.

Nondefective Matrices with
Complex Eigenvalues
If all of the eigenvectors of A, real and complex, are
linearly independent, then a fundamental set of real
solutions of x' = Ax consists of solutions of the form
above associated with complex eigenvalues and
solutions of the form eλjtvj associated with real
eigenvalues.
 For example, suppose that λ1 = μ + iν, λ2 = μ − iν, and
that λ3, . . . , λn are all real and distinct. Let the
corresponding eigenvectors be v1 = a + ib, v2 = a −
ib, v3, . . . , vn. Then the general solution of x' = Ax is
x = c1x1(t) + c2x2(t) + c3eλ3tv3 + ・・・ + cneλnt vn,
where x1(t) and x2(t) are given by above equations.
 We emphasize that this analysis applies only if the
coefficient matrix A is real, for it is only then that
complex eigenvalues and eigenvectors occur in
conjugate pairs.

Example
Answer: The general solution is
Plots of the components and
A typical solution trajectory
Natural Frequencies and Principal
Modes of Vibration.
Consider again the system of two masses
and three springs shown in Figure.
If we assume that there are no external forces,
then F1(t) = 0, F2(t) = 0 and from Section 6.1
we get the homogeneous system
Natural Frequencies and Principal
Modes of Vibration (Ctd.)



This system is said to have two degrees of
freedom since there are two independent
coordinates, x1 and x2, necessary to describe the
motion of the system.
If both masses will undergo harmonic motion at the
same frequency, the frequency is called a natural
frequency of the system, and the motion is called
a principal mode of vibration.
The principal mode of vibration corresponding to
the lowest natural frequency is referred to as the
first mode. The principal mode of vibration
corresponding to the next higher frequency is
called the second mode and so on.
Example
Answer: The general solution for arbitrary constants c1,
c2, c3, and c4 is
The first principal mode of vibration

The motions of
m1 and m2 are in
phase but have
different
amplitudes.
The second principal mode of
vibration

The
motions of
m1 and m2
are 180
degrees out
of phase
and have
different
amplitudes.
Component plots
Component
plots of the
solution
corresponding
to the initial
condition x(0)
= (0, 2, 0, 0)T