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Differential Equations
Chapter 03:
Systems of Two
First Order
Equations
Brannan
Copyright © 2010 by John Wiley & Sons, Inc.
All rights reserved.
Chapter 3 Systems of Two First
Order Equations



We introduce systems of two first order equations
In this chapter, we consider only systems of two
first order equations and we focus most of our
attention on systems of the simplest kind: two first
order linear equations with constant coefficients.
Our goals are to show what kinds of solutions such
a system may have and how the solutions can be
determined and displayed graphically, so that they
can be easily visualized.
Chapter 3
Systems of Two First Order
Equations







3.1 Systems of Two Linear Algebraic Equations
3.2 Systems of Two First Order Linear Differential
Equations
3.3 Homogeneous Linear Systems with Constant
Coefficients
3.4 Complex Eigenvalues
3.5 Repeated Eigenvalues
3.6 A Brief Introduction to Nonlinear Systems
3.7 Numerical Methods for Systems of First
Order Equations
3.1 Systems of Two Linear Algebraic
Equations

Solutions to a system of equations
There are three distinct possibilities for
two straight lines in a plane: they may
intersect at a single point, they may be
parallel and nonintersecting, or they
may be coincident.
Examples:
 1. 3x1 − x2 = 8, x1 + 2x2 = 5.
 2. x1 + 2x2 = 1, x1 + 2x2 = 5.
 3. 2x1 + 4x2 = 10, x1 + 2x2 = 5.

Cramer’s Rule – THEOREM 3.1.1
The system
a11x1 + a12x2 = b1,
a21x1 + a22x2 = b2,
has a unique solution if and only if the
determinant Δ = a11a22 − a12a21 = 0.
The solution is given by
If Δ = 0, then the system has either no solution
or infinitely many.
Matrix Method

a11 a12 

Consider coefficient matrix, A 
a

a
 21 22 
If A−1 exists, then A is called nonsingular or
invertible. On the other hand, if A−1 does
not exist, then A is said to be singular or
noninvertible.
b1 

 The solution to Ax=B is x = A−1b. b   

b2 
Homogeneous System
THEOREM 3.1.2 The homogeneous system
Ax = 0 always has the trivial solution x1 =
0, x2 = 0, and this is the only solution when
det(A) ≠ 0. Nontrivial solutions exist if and
only if det(A) =0. In this case, unless A = 0,
all solutions are proportional to any
nontrivial solution; in other words, they lie
on a line through the origin. If A = 0, then
every point in the x1x2-plane is a solution of
system.
Example: Solve the system
3x1 − x2 = 0, x1 + 2x2 = 0.
Eigenvalues and Eigenvectors
Eigenvalues (λ) of the matrix A are the
solutions to Ax = λx. The eigenvector x
corresponding to the eigenvalue λ is obtained
by solving Ax = λx for x for the given λ.
 For a 2X2 matrix Ax = λx reduces to

a12 
 a11  

 x  0 Since det(A-λI)=0, get
a22   
 a21
Characteristic equation
The characteristic equation of the
matrix A is
λ2 − (a11 + a22)λ + a11a22 − a12a21 = 0.
Solutions determine the eigenvalues.
The two solutions, the eigenvalues λ1
and λ2, may be real and different, real
and equal, or complex conjugates.
Examples
Find the eigenvalues and eigenvectors of the
matrix A.
 1.

2.

3.
THEOREM 3.1.3

Let A have real or complex eigenvalues λ1 and
λ2 such that λ1≠λ2, and let the corresponding
eigenvectors be x1 and x2. If X is the matrix
with first and second columns taken to be x1
and x2, respectively, then det(X) ≠0.
That is,
3.2 Systems of Two First Order
Linear Differential Equations
du/dt = Ku + b.
where K is a given 2X2 matrix and b a
given 2x1 matrix. U is 2X1 matrix of
unknowns whose first derivative is
du/dt. We solve this system subject to
a given initial condition u(0)= u0, a
2X1 matrix with given values.
Example
Here
Terminology
The components of u are scalar valued
functions of t, so we can plot their graphs.
Plots of u1 and u2 versus t are called
component plots.
 The variables u1 and u2 are often called
state variables, since their values at any
time describe the state of the system.
 Similarly, the vector u = u1i + u2j is called
the state vector of the system. The u1u2plane itself is called the state space. If
there are only two state variables, the u1u2plane may be called the state plane or,
more commonly, the phase plane.

Direction Fields

The right side of a system of first order
equations du/dt = Ku + b defines a vector
field that governs the direction and speed of
motion of the solution at each point in the
phase plane. Because the vectors
generated by a vector field for a specific
system often vary significantly in length, it is
customary to scale each nonzero vector so
that they all have the same length. These
vectors are then referred to as direction
field vectors for the system and the
resulting picture is called the direction
field.
Phase Portraits

Using a computer we can to generate
solution trajectories. A plot of a
representative sample of the
trajectories, including any constant
solutions, is called a phase portrait of
the system of equations.
Phase portrait for the example
General Solutions of Two First
Order Linear Equations
THEOREM 3.2.1 Existence and Uniqueness
of Solutions
Let each of the functions p11, . . . , p22, g1, and g2 be
continuous on an open interval I =α < t < β, let t0 be
any point in I, and let x0 and y0 be any given
numbers. Then there exists a unique solution of the
system
that also satisfies the initial conditions
Further, the solution exists throughout the interval I.
First Order Linear Equations

In matrix form, we write

The system above is called a first order
linear system of dimension two because it
consists of first order equations and
because its state space (the xy-plane) is
two-dimensional. Further, if g(t) = 0 for all t,
that is, g1(t) = g2(t) = 0 for all t, then the
system is said to be homogeneous.
Otherwise, it is nonhomogeneous.
Linear Autonomous Systems
If the right side of
does not depend explicitly on the independent
variable t, the system is said to be
autonomous.
Then the coefficient matrix P and the
components of the vector g must be
constants. We use the notation
dx/dt = Ax + b, where A is a constant
matrix and b is a constant vector, to denote
autonomous linear systems.
Critical points of linear
autonomous system
For the linear autonomous system, we find
the equilibrium solutions, or critical
points, by setting dx/dt equal to zero.
Hence any solution of Ax = −b is a critical
point of the system.
If the coefficient matrix A has an inverse, as
we usually assume, then Ax = −b has a
single solution, namely, x=−A−1b.
This is then the only critical point of the
system. However, if A is singular, then
Ax = −b has either no solution or infinitely
many.
Transformation of a Second Order
Equation to a System of First
Order Equations
Consider the second order equation
y'' + p(t)y' + q(t)y = g(t),
where p, q, and g are given functions that we
assume to be continuous on an interval I.
Substituting x1 = y and x2 = y'. This system can
be transformed to a system of two first order
equations,
Example
Consider the differential equation
u'' + 0.25u' + 2u = 3 sin t. Suppose that initial
conditions u(0) = 2, u(0) = −2. Transform this
problem into an equivalent one for a system
of first order equations. Write the matrix
notation for this initial value problem.
Answer:
Component plots of the solution to
this initial value problem
3.3 Homogeneous Linear Systems
with Constant Coefficients
Reducing x' = Ax + b to x' = Ax
If A has an inverse, then the only critical, or
equilibrium, point of x' = Ax + b is xeq =
−A−1b. In such cases it is convenient to shift
the origin of the phase plane to the critical
point using the coordinate transformation x
= xeq + x˜. Substituting, we get dx˜/dt = Ax˜.
Therefore, if x = φ(t) is a solution of the
homogeneous system x' = Ax, then the
solution of the nonhomogeneous system x'
= Ax + b is given by x = φ(t) + xeq = φ(t) −
A−1b.

The Eigenvalue Method for Solving
x' = Ax
Consider a general system of two first
order linear homogeneous differential
equations with constant coefficients
dx/dt=Ax.
 x = eλtv is a solution of dx/dt = Ax
provided that λ is an eigenvalue and v
is a corresponding eigenvector of the
coefficient matrix A.Hence Av = λv, or
(A − λI)v = 0.

THEOREM 3.3.1 - Principle of
Superposition
Suppose that x1(t) = eλ1tv1 and x2(t) =
eλ2tv2 are solutions of
dx/dt = Ax.
Then the expression
x = c1x1(t) + c2x2(t),
where c1 and c2 are arbitrary constants,
is also a solution. We assume that λ1
and λ2 are real and different.
Example
 1 0 
 x.
Consider the system dx/dt = 
 0  4
Find solutions of the system and then find the
particular solution that satisfies the initial
Condition
 2
x(0) =   .
 3
Answer:
1
 4t  0 
x  2e    3e  
 0
1
t
Wronskian determinant
The determinant
is called the Wronskian determinant or, more simply,
the Wronskian of the two vectors x1 and x2. If x1(t) =
eλ1tv1 and x2(t) = eλ2tv2, then their Wronskian is
Two solutions x1(t) and x2(t) of whose Wronskian is not
zero are referred to as a fundamental set of
solutions. The linear combination of x1 and x2 given
with arbitrary coefficients c1 and c2, x = c1x1(t) +
c2x2(t), is called the general solution.
THEOREM 3.3.2
Suppose that x1(t) and x2(t) are two solutions
of dx/dt = Ax, and that their Wronskian is
not zero. Then x1(t) and x2(t) form a
fundamental set of solutions, and the
general solution is given by, x = c1x1(t) +
c2x2(t), where c1 and c2 are arbitrary
constants. If there is a given initial condition
x(t0) = x0, where x0 is any constant vector,
then this condition determines the constants
c1 and c2 uniquely.
Note: The theorem is true if coefficient matrix A has eigenvalues
that are real and different. It is also valid even when the
eigenvalues are complex or repeated.
EXAMPLE - A Rockbed
Heat Storage System Revisited
Consider again the greenhouse/rockbed heat
storage problem with coordinates centered
at the critical point given by,
 13 3 


4  x = Ax.
dx/dt =  8
1
 1
 
4
 4
Find the general solution of this system. Then
plot a direction field, a phase portrait, and
several component plots of the system.
Answer

The general solution is

The eigenvalues are λ1 = -7/4 and λ2 = −1/8.
Direction field and phase portrait for the
system is shown in the next slide.
Nodal Sources and Nodal Sinks
The pattern of
trajectories in
Figure is typical
of all second order
systems x' = Ax
whose eigenvalues
are real, different,
and of the same
sign. The origin
is called a node for
such a system.
Nodal Sources and Nodal Sinks
If the eigenvalues were positive rather than
negative, then the trajectories would be
similar but traversed in the outward
direction.
 Nodes are asymptotically stable if the
eigenvalues are negative and unstable if
the eigenvalues are positive.
 Asymptotically stable nodes and unstable
nodes are also referred to as nodal sinks
and nodal sources respectively.

Example
Consider the system
 1 1
dx/dt = 
x = Ax.

 4 1


Find the general solution and draw a phase
portrait.
Answer

The general solution is

The eigenvalues are λ1 = 3 and λ2 = −1.
Direction field and phase portrait for the
system is shown in the next slide.
Saddle Points
The pattern of
trajectories in Figure
is typical of all
second order
systems x' = Ax for
which the
eigenvalues are real
and of opposite
signs. The origin is
called a saddle point
in this case. Saddle
points are always
unstable because
almost all trajectories
depart from them as t
increases.
3.4 Complex Eigenvalues
Consider a two dimensional system x= Ax with complex
conjugate eigenvalues
To solve the system, find the eigenvalues and
eigenvectors, observing that they are complex
conjugates. Then write down x1(t) and separate it into its
real and imaginary parts u(t) and w(t), respectively.
Finally, form a linear combination of u(t) and w(t), x =
c1u(t) + c2w(t).
Of course, if complex-valued solutions are acceptable,
you can simply use the solutions x1(t) and x2(t).
Thus Theorem 3.3.2 is also valid when the eigenvalues
are complex.
Example
Q: Consider the system
Find a fundamental set of solutions and
display them graphically in a phase portrait
and component plots.
A: The General solution

Component plots for the solutions
u(t) and w(t) of the system
A direction field and phase portrait
for the system
Spiral Points

The phase portrait in previous Figure is typical of
all two-dimensional systems x' = Ax whose
eigenvalues are complex with a negative real part.
The origin is called a spiral point and is
asymptotically stable because all trajectories
approach it as t increases. Such a spiral point is
often called a spiral sink. For a system whose
eigenvalues have a positive real part, the
trajectories are similar to those in Figure, but the
direction of motion is away from the origin and the
trajectories become unbounded. In this case, the
origin is unstable and is often called a spiral
source.
Centers
If the real part of the eigenvalues is
zero, then there is no exponential
factor in the solution and the
trajectories neither approach the
origin nor become unbounded.
Instead, they repeatedly traverse
a closed curve about the origin.
An example of this behavior can be
seen in Figure to left. In this case,
the origin is called a center and
is said to be stable, but not
asymptotically stable. In all three
cases, the direction of motion
may be either clockwise, as in
previous Example, or
counterclockwise, depending on
the elements of the coefficient
matrix A.
Summary
For two-dimensional systems with real
coefficients, we have now completed our
description of the three main cases that
can occur:
1. Eigenvalues are real and have opposite
signs; x = 0 is a saddle point.
2. Eigenvalues are real and have the same
sign but are unequal; x = 0 is a node.
3. Eigenvalues are complex with nonzero real
part; x = 0 is a spiral point.
3.4 Repeated Eigenvalues
An Example
Q: Consider the system x' = Ax, where
Draw a direction field, a phase portrait, and
typical component plots.
A: The eigenvalues are λ1 = λ2 = −1. General
solution x = c1x1(t) + c2x2(t) where
Typical component plots for the
system
A direction field and phase portrait
for the system
Proper node or Star Point
It is possible to show that the only 2 X2
matrices with a repeated eigenvalue and
two independent eigenvectors are the
diagonal matrices with the eigenvalues
along the diagonal. Such matrices form a
rather special class, since each of them is
proportional to the identity matrix. The
system in above Example is entirely typical
of this class of systems.
 In this case the origin is called a proper
node or, sometimes, a star point.

Repeated Eigenvalues (in general)
Consider two-dimensional linear
homogeneous systems with constant
coefficients x' = Ax.
 Suppose that λ1 is a repeated
eigenvalue of the matrix A and that
there is only one independent
eigenvector v1. Then one solution is
x1(t) = e λ1t v1. A second solution is
x2(t) = teλ1tv1 + eλ1tw, where w
satisfies (A − λ1I)w = v1.

Repeated Eigenvalues (Ctd.)
The vector w is called a generalized
eigenvector corresponding to the
eigenvalue λ1.
 In the case where the 2X2 matrix A
has a repeated eigenvalue and only
one eigenvector, the origin is called an
improper or degenerate node.

Example
Q: Consider the system
Find the eigenvalues and eigenvectors of the
coefficient matrix, and then find the
generalsolution of the system. Draw a
direction field, phase portrait, and component
plots.
A: The eigenvalues are λ1 = λ2 = −1/2. General
solution x = c1x1(t) + c2x2(t) where
A direction field and phase portrait
for the system
improper or
degenerate node
Typical plots of x1 versus t for the
system
Summary of Results
3.6 A Brief Introduction to
Nonlinear Systems

In Section 3.2, we introduced the general twodimensional first order linear system

Of course, two-dimensional systems that are not of the
form (1) or (2) may also occur. Such systems are said to
be nonlinear.
THEOREM 3.6.1 - Existence and
Uniqueness of Solutions.
Let each of the functions f and g and the partial
derivatives ∂f /∂x, ∂f /∂y, ∂g/∂x, and ∂g/∂y be
continuous in a region R of txy-space defined
by α < t < β, α1 < x < β1, α2 < y < β2, and let the
point (t0, x0, y0) be in R. Then there is an
interval |t − t0| < h in which there exists a
unique solution of the system of differential
equations
that also satisfies the initial conditions x(t0) = x0,
y(t0) = y0.
Autonomous Systems



It is usually impossible to solve nonlinear systems
exactly by analytical methods.
Therefore for such systems graphical methods and
numerical approximations become even more
important. In the next section, we will extend our
discussion of approximate numerical methods to twodimensional systems. Here we will consider systems
for which direction fields and phase portraits are of
particular importance. These are systems that do not
depend explicitly on the independent variable t. In
other words, the functions f and g in the equation
depend only on x and y and not on t.
Such a system is called autonomous, and can be
written in the form
Equilibrium points or Critical
points
To find equilibrium, or constant, solutions of
the autonomous system, we set dx/dt and
dy/dt equal to zero, and solve the resulting
equations
f (x, y) = 0, g(x, y) = 0
for x and y. Any solution of these is a point in
the phase plane that is a trajectory of an
equilibrium solution. Such points are called
equilibrium points or critical points.
 Depending on the particular forms of f and
g, the nonlinear system can have any
number of critical points, ranging from none
to infinitely many.

Example
Consider the system
dx/dt = x − y,
dy/dt= 2x − y − x2.
 Find a function H(x, y) such that the
trajectories of the system lie on the
level curves of H. Find the critical
points and draw a phase portrait for
the given system. Describe the
behavior of its trajectories.

Example (Ctd.)
To find the critical points, solve the
equations x − y = 0, 2x − y − x2 = 0.
The critical points are (0, 0) and (1, 1).
 To determine the trajectories, note
that for this system, becomes
dy/dx =(2x − y − x2)/(x − y)
This is exact and so solutions satisfy
H(x, y) = x2 − xy + 1/2 y2 − 1/3 x3 = c,
where c is an arbitrary constant.

A phase portrait for the system
3.7 Numerical Methods for
Systems of First Order Equations
Numerical methods for approximating the solutions of
initial value problems for a single first order differential
equation In Sections 1.3, 2.7, and 2.8 can be used.
 The algorithms are the same for nonlinear and for linear
equations, so we will not restrict ourselves to linear
equations in this section. We consider a system of two
first order equations
x' = f (t, x, y), y' = g(t, x, y),
with the initial conditions x(t0) = x0, y(t0) = y0.
The functions f and g are assumed to satisfy the
conditions of Theorem 3.6.1 so that the initial value
problem above has a unique solution in some interval
of the t-axis containing the point t0. We wish to
determine approximate values x1, x2, . . . , xn, . . . and
y1, y2, . . . , yn, . . . of the solution x = φ(t), y = ψ(t) at the
points tn = t0 + nh with n = 1, 2, . . . .

Euler formula

The scalar Euler formula tn+1 = tn + h, xn+1
= xn + h fn is replaced by
Runge–Kutta method

The Runge–Kutta method can be extended
to a system. For the step from tn to tn+1 we
have
Example
Determine approximate values of the solution
x=φ(t), y=ψ(t) of the initial value problem
x' = −x + 4y, y' = x − y, x(0) = 2, y(0) = −0.5,
at the point t = 0.2. Use the Euler method with
h = 0.1 and the Runge–Kutta method with h
= 0.2. Compare the results with the values
of the exact solution:
φ(t) =(et + 3e−3t)/2, ψ(t) =(et − 3e−3t)/4
Approximations to the solution of the initial
value problem using the Euler method (h =
0.1) and the Runge–Kutta method (h = 0.2).
Summary
Section 3.1 Two-Dimensional
Linear Algebra

1.
2.

Matrix notation for a linear algebraic system of
two equations in two unknowns is Ax = b.
If det A ≠ 0, the unique solution of Ax = b is x =
A−1b.
If det A = 0, Ax = b may have (i) no solution, or (ii)
a straight line of solutions in the plane; in
particular, if b = 0 and A ≠ 0, the solution set is a
straight line passing through the origin.
The eigenvalue problem: (A − λI)x = 0. The
eigenvalues of A are solutions of the
characteristic equation det(A − λI) = 0. An
eigenvector for the eigenvalue λ is a nonzero
solution of (A − λI)x = 0. Eigenvalues may be real
and different, real and equal, or complex
conjugates.
Section 3.2 Systems of Two First
Order Linear Equations
Section 3.3 Homogeneous
Systems with Constant
Coefficients: x' = Ax
Section 3.4 Complex Eigenvalues
If the eigenvalues of A are μ±iν, ν ≠ 0,
with corresponding eigenvectors a±ib,
a fundamental set of real vector
solutions of x = Ax consists of
Re{exp[(μ + iν)t][a + ib]} = exp(μt)(cos
νta − sin νtb) and Im{exp[(μ + iν)t][a +
ib]} = exp(μt)(sin νta + cos νtb).
 If μ ≠ 0, then the critical point (the
origin) is a spiral point. If μ = 0, then
the critical point is a center.

Section 3.5 Repeated Eigenvalues
If A has a single repeated eigenvalue λ,
then a general solution of x' = Ax is
(i) x = c1eλtv1 + c2eλtv2 if v1 and v2 are
independent eigenvectors, or
(ii) x = c1eλtv + c2eλt (w + tv), where (A − λI)w
= v if v is the only eigenvector of A.
 The critical point at the origin is a proper
node if there are two independent
eigenvectors, and an improper or
degenerate node if there is only one
eigenvector.

Section 3.6 Nonlinear Systems
Nonautonomous: x' = f (t, x),
 Autonomous: x ' = f (x)
 Theorem 3.6.1 provides conditions that
guarantee, locally in time, existence and
uniqueness of solutions to the initial value
problem x' =f(t, x), x(t0) = x0.
 Examples of two-dimensional nonlinear
autonomous systems suggest that locally
their solutions behave much like solutions
of linear systems.

Section 3.7 Numerical
Approximation Methods for
Systems

The Euler and Runge–Kutta methods
described in Chapters 1 and 2 are
extended to systems of first order
equations, and are illustrated for a
typical two-dimensional system.