Topic 09: Linear System of First Order ODEs
Second Part of MA102 Mathematics II: Winter Semester of AY 2024-2025
MGPP, SN, SHB: IIT Guwahati
Sections 11.1 to 11.7, 7.3, 7.4, 7.6, 7.7
of
S. L. Ross, Differential Equations, 3rd Edition, Wiley
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System of ordinary linear differential equations, Lecture notes of Differential Equations
Discusses the system of linear differential equations, in details, taking them in matrix form and performing matrix algebra
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Topic 09: Linear System of First Order ODEs
Second Part of MA102 Mathematics II: Winter Semester of AY 2024-
MGPP, SN, SHB: IIT Guwahati
Sections 11.1 to 11.7, 7.3, 7.4, 7.6, 7.
of
S. L. Ross, Differential Equations, 3rd Edition, Wiley
Linear System of First Order ODEs
Text:
I (^) S. L. Ross, Differential Equations, Third Edition, Wiley.
References:
I Lawrence Perko, Differential Equations and Dynamical Systems, Third Edition,
Springer.
I G. F. Simmons & S. G. Krantz, Differential Equations: Theory, Technique, and
Practice, Tata McGraw-Hill.
I Dennis G. Zill, Differential Equations, Cengage Learning, India Edition.
Refer Linear Systems of First Order ODEs:
- Ross Book Sections: 11.1, 11.2, 11.3, 11.4, 11.5, 11.6, 11.7, 7.3, 7.4, 7.6, 7.7.
- Perko Book Sections: 1.2, 1.3, 1.4, 1.6, 1.7.
- Zill Book Sections: 8.1, 8.2, 8.3, 8.4.
System of First Order ODEs
There are many physical problems, more than one element/ object interact with each
other in some manner. To mathematically model such physical problems, we need a
system of two or more differential equations. For example,
I (^) Predator-Prey Model
I Mechanics: Motion of certain Spring-Mass System (Two masses move on
fricrtionless surface)
I Electrical Circuit: Parallel LRC Circuit / more than one loop
I (^) Mixture Problems: Connected Mixing Tanks
Lotka-Volterra Predator-Prey Model
Two different species, say, Foxes and Rabbits interact with in same ecosystem. A
simple model supposes that rabbits eat only grass and foxes eat only rabbits. In other
words, fox is a predator and rabbit is a prey. Let F(t) and R(t) denote the population of
foxes and rabbits respectively, at time t.
If there were no rabbits then foxes won’t have food to eat and hence the population of
foxes would decline in number according to
dF
dt
= −aF where a > 0.
When rabbits are present, the number of interactions/ encounters between these two
species per unit time is proportional to their populations F and R, that is, proportional
to the product FR. Thus when rabbits are present, there is a supply of food to foxes, so
foxes are added to the system at the rate of bFR, b > 0. Therefore
dF
dt
= −aF + bFR where a > 0 and b > 0.
Competition Model
Suppose two different species of animals occupy the same ecosystem, not as predator
and prey, but rather as competitors for the same resources (such as food and living
space) in the system.
In absence of the other, let us assume that the rate at which each population grows is
given by
dx
dt
= ax and
dy
dt
= cy where a > 0 , c > 0.
Since the two species compete, another assumption might be that each of these rates
is diminished simply by the influence or existence of other population. Thus, this model
leads to the following linear system of ODEs. Competition Model
dx
dt
= a x − b y
dy
dt
= c y − d x
where a, b, c and d are positive real constants.
Connected Mixing Tanks
Consider the two tanks A and B as shown in Figure. Let us suppose that tank A
contains 50 gallons of water in which 25 pounds of salt is dissolved. Suppose tank B
contains 50 gallons of pure water. Liquid is pumped into and out of the tanks as
indicated in the Figure. The mixture exchanged between the two tanks and the liquid
pumped out of tank B are assumed to be well stirred. Construct a mathematical model
that describes the number of pounds x 1
(t) and x 2
(t) of salt in tanks A and B,
respectively, at time t.
dx 1
dt
x 1
x 2
dx 2
dt
x 1
x 2
with the initial conditions x 1
(0) = 25 and x 2
Linear System of First Order ODEs
A Linear system of first order ODEs (in normal form) is given by
dx 1
dt
= a 11 (t)x 1 + · · · + a 1 n(t)xn + b 1 (t)
dx 2
dt
= a 21
(t)x 1
- · · · + a 2 n
(t)x n
- b 2
(t)
dx n
dt
= a n 1
(t)x 1
- · · · + a nn
(t)x n
- b n
(t)
It can be written in the vector differential equation (VDE) form as
dx
dt
= A x + f or
dX
dt
= A X + F or
d
X
dt
A
X +
F ,
A = A(t) =
a 11
(t) · · · a 1 n
(t)
a n 1
(t) · · · a nn
(t)
, x = x(t) =
x 1
(t)
x n
(t)
and f = f(t) =
b 1
(t)
b n
(t)
.
For writing purpose, we prefer the capital letters (instead of bold-face letters) to denote
the Vector Quantities.
So, while writing on paper, one can prefer to use the notation
dX
dt
= A X + F , (1)
where A = A(t) =
a 11
(t) · · · a 1 n
(t)
a n 1
(t) · · · a nn
(t)
,
X = X(t) =
x 1
(t)
x n
(t)
and F = F(t) =
b 1
(t)
b n
(t)
.
Definition: If F ≡ 0 in Equation (1), then the VDE X
′ = AX is called the homogeneous
system. If F. 0 in Equation (1), then the VDE X
′ = AX + F is called the
nonhomogeneous system.
Examples
Example 3: Here y(x) = (y 1
(x), y 2
(x), y 3
(x)). That is, x is independent variable and y 1
,
y 2
, y 3
are dependent variables.
dy 1
dx
= x
2
y 1
(x + 1) y 2
3 y 3
x
3
dy 2
dx
= xe
x
y 1
- x
3
y 2
- 8 y 3
− e
x
dy 3
dx
= x y 1
- e
x
y 2
5 y 3
x
Example 4:
dx
dt
= A x + f ,
where x =
x 1
x 2
x 3
, A =
t
2 (t + 1) 5
2 t
3
e
t
e
−t − 4 cos t
and f =
3 t
3
−e
t
− 8 e
t
.
Solution to the Linear System of First Order ODEs
Definition 1
By a solution of the vector differential equation
dx
dt
= A x + f ,
we mean n × 1 column vector function
Φ(t) =
φ 1
(t)
φ n
(t)
whose components φ 1
, φ 2
,.. ., φ n
each have a continuous derivative on the real
interval a ≤ t ≤ b, which satisfies that
dΦ(t)
dt
= A(t) Φ(t) + f(t) for all t ∈ [a, b].
Homogeneous Linear System of First Order ODEs
A Homogeneous Linear system of first order ODEs is given by
dx 1
dt
= a 11 (t)x 1 + · · · + a 1 n(t)xn
dx 2
dt
= a 21
(t)x 1
- · · · + a 2 n
(t)x n
dx n
dt
= a n 1
(t)x 1
- · · · + a nn
(t)x n
It can be written in the vector differential equation (VDE) form as
dx
dt
= A x or
dX
dt
= A X or
d
X
dt
A
X ,
A = A(t) =
a 11
(t) · · · a 1 n
(t)
a n 1
(t) · · · a nn
(t)
and x = x(t) =
x 1
(t)
x n
(t)
.
Example 3
Consider the homogeneous linear VDE x
′
(t) = A(t) x(t) where
A(t) =
and x(t) =
x 1
(t)
x 2
(t)
x 3
(t)
Then the column vector function
Φ(t) =
e
3 t
− 2 e
3 t
−e
3 t
is a solution of the above given VDE on every real interval a ≤ t ≤ b.
Checking Φ
′ (t) = A(t) Φ(t):
3 e
3 t
− 6 e
3 t
− 3 e
3 t
e
3 t
− 2 e
3 t
−e
3 t
for all t ∈ [a, b].
Outline of Proof for E & U of Solution to the IVP-HLVDE
Step 1:
The solution to the IVP (3) is equivalent to the solution of the vector integral equation
x(t) = x 0
t
t 0
A(s) x(s) ds (4)
which gives x(t 0
) = x 0
.
Step 2: Picard Iterates (For the case t ∈ I with t ≤ t 0
, proof is similar).
Define the Picard iterates as follows:
x 0
(t) = x 0
x n+ 1
(t) = x 0
(t) +
t
t 0
A(s) x n
(s) ds t ≥ t 0
, t ∈ I (5)
for n = 0 , 1 , 2 , 3 ,.. .. Then, the sequence {xn(t)} is well-defined and it is a Cauchy
sequence in R
n for each t. Therefore, the sequence {x n
(t)} converges uniformly on
[a, b] to a function x(t) (say).
Continuation of the Previous Slide
Since x n
(t) → x(t) uniformly on I, we can take the limit under the integral sign of (5) to
get
x(t) = x 0
(t) +
t
t 0
A(s) x(s) ds
which proves that x(t) is the solution of the integral equation (4) and hence x(t) is a
solution to the IVP (3).
Step 3: Uniqueness of the Solution
If x(t) and y(t) are the solutions of (3) then
x(t) − y(t) =
t
t 0
A(s)(x(s) − y(s)) ds =⇒ |x(t) − y(t)| ≤ M
t
t 0
|x(s) − y(s)| ds. Thus for any
given > 0 , we get from the above inequality, |x(t) − y(t)| < + M
t
t 0
|x(s) − y(s)| ds. By
Grownwall integral inequality, one can get
|x(t) − y(t)| < exp(M(t − t 0
)) for t ≥ t 0
Since the above inequality is true for each > 0 , we can get |x(t) − y(t)| = 0 and hence
x(t) = y(t) for t ≥ t 0
.