This document provides an overview of Topic 8 on Ordinary Differential Equations (ODEs) for the course SE301 Numerical Methods. It introduces ODEs and discusses how to classify them by order, linearity, and type of auxiliary conditions. Numerical methods are needed to solve most ODEs and work by approximating the Taylor series expansion. The topic will cover Taylor series methods, Runge-Kutta methods, solving systems of ODEs, and boundary value problems. Lesson 1 introduces basic definitions of ODEs, how to classify them, and discusses analytical versus numerical solutions.

1. CISE301_Topic8L1 1
SE301: Numerical Methods
Topic 8
Ordinary Differential Equations (ODEs)
Lecture 28-36
KFUPM
(Term 101)
Section 04
Read 25.1-25.4, 26-2, 27-1

2. CISE301_Topic8L1 2
Objectives of Topic 8
 Solve Ordinary Differential Equations
(ODEs).
 Appreciate the importance of numerical
methods in solving ODEs.
 Assess the reliability of the different
techniques.
 Select the appropriate method for any
particular problem.

3. CISE301_Topic8L1 3
Outline of Topic 8
 Lesson 1: Introduction to ODEs
 Lesson 2: Taylor series methods
 Lesson 3: Midpoint and Heun’s method
 Lessons 4-5: Runge-Kutta methods
 Lesson 6: Solving systems of ODEs
 Lesson 7: Multiple step Methods
 Lesson 8-9: Boundary value Problems

4. CISE301_Topic8L1 4
Lecture 28
Lesson 1: Introduction to ODEs

5. CISE301_Topic8L1 5
Learning Objectives of Lesson 1
 Recall basic definitions of ODEs:
 Order
 Linearity
 Initial conditions
 Solution
 Classify ODEs based on:
 Order, linearity, and conditions.
 Classify the solution methods.

6. CISE301_Topic8L1 6
Partial Derivatives
u is a function of
more than one
independent variable
Ordinary Derivatives
v is a function of one
independent variable
Derivatives
Derivatives
dt
dv
y
u



7. CISE301_Topic8L1 7
Partial Differential Equations
involve one or more
partial derivatives of
unknown functions
Ordinary Differential Equations
involve one or more
Ordinary derivatives of
unknown functions
Differential Equations
Differential
Equations
1
6
2
2

 tv
dt
v
d
0
2
2
2
2






x
u
y
u

8. CISE301_Topic8L1 8
Ordinary Differential Equations
)
cos(
)
(
2
)
(
5
)
(
)
(
)
(
:
2
2
t
t
x
dt
t
dx
dt
t
x
d
e
t
v
dt
t
dv
Examples
t





Ordinary Differential Equations (ODEs) involve one or
more ordinary derivatives of unknown functions with
respect to one independent variable
x(t): unknown function
t: independent variable

9. CISE301_Topic8L1 9
Example of ODE:
Model of Falling Parachutist
The velocity of a falling
parachutist is given by:
velocity
v
t
coefficien
drag
c
mass
M
v
M
c
t
d
v
d
:
:
:
8
.
9 


10. CISE301_Topic8L1 10
Definitions
v
M
c
dt
dv

 8
.
9
Ordinary
differential
equation

11. CISE301_Topic8L1 11
Definitions (Cont.)
v
M
c
t
d
v
d

 8
.
9 (Dependent
variable)
unknown
function to be
determined

12. CISE301_Topic8L1 12
Definitions (Cont.)
v
M
c
t
d
v
d

 8
.
9
(independent variable)
the variable with respect to which
other variables are differentiated

13. CISE301_Topic8L1 13
Order of a Differential Equation
1
)
(
2
)
(
)
(
)
cos(
)
(
2
)
(
5
)
(
)
(
)
(
:
4
3
2
2
2
2
















t
x
dt
t
dx
dt
t
x
d
t
t
x
dt
t
dx
dt
t
x
d
e
t
x
dt
t
dx
Examples
t
The order of an ordinary differential equation is the order
of the highest order derivative.
Second order ODE
First order ODE
Second order ODE

14. CISE301_Topic8L1 14
A solution to a differential equation is a function that
satisfies the equation.
Solution of a Differential Equation
0
)
(
)
(
:

 t
x
dt
t
dx
Example
0
)
(
)
(
)
(
:
Proof
)
(












t
t
t
t
e
e
t
x
dt
t
dx
e
dt
t
dx
e
t
x
Solution

15. CISE301_Topic8L1 15
Linear ODE
1
)
(
)
(
)
(
)
cos(
)
(
2
)
(
5
)
(
)
(
)
(
:
3
2
2
2
2
2
















t
x
dt
t
dx
dt
t
x
d
t
t
x
t
dt
t
dx
dt
t
x
d
e
t
x
dt
t
dx
Examples
t
An ODE is linear if
The unknown function and its derivatives appear to power one
No product of the unknown function and/or its derivatives
Linear ODE
Linear ODE
Non-linear ODE

16. CISE301_Topic8L1 16
Nonlinear ODE
1
)
(
)
(
)
(
2
)
(
)
(
5
)
(
1
))
(
cos(
)
(
:
ODE
nonlinear
of
Examples
2
2
2
2







t
x
dt
t
dx
dt
t
x
d
t
x
dt
t
dx
dt
t
x
d
t
x
dt
t
dx
An ODE is linear if
The unknown function and its derivatives appear to power one
No product of the unknown function and/or its derivatives

17. CISE301_Topic8L1 17
Solutions of Ordinary Differential
Equations
0
)
(
4
)
(
ODE
the
to
solution
a
is
)
2
cos(
)
(
2
2



t
x
dt
t
x
d
t
t
x
Is it unique?
solutions.
are
constant)
real
a
is
(where
)
2
cos(
)
(
form
the
of
functions
All
c
c
t
t
x 


18. CISE301_Topic8L1 18
Uniqueness of a Solution
b
x
a
x
t
x
dt
t
x
d




)
0
(
)
0
(
0
)
(
4
)
(
2
2

In order to uniquely specify a solution to an nth
order differential equation we need n conditions.
Second order ODE
Two conditions are
needed to uniquely
specify the solution

19. CISE301_Topic8L1 19
Auxiliary Conditions
Boundary Conditions
 The conditions are not at
one point of the
independent variable
Initial Conditions
 All conditions are at one
point of the independent
variable
Auxiliary Conditions

20. CISE301_Topic8L1 20
Boundary-Value and
Initial value Problems
Boundary-Value Problems
 The auxiliary conditions are
not at one point of the
independent variable
 More difficult to solve than
initial value problems
5
.
1
)
2
(
,
1
)
0
(
2 2




 
x
x
e
x
x
x t



Initial-Value Problems
 The auxiliary conditions
are at one point of the
independent
variable
5
.
2
)
0
(
,
1
)
0
(
2 2




 
x
x
e
x
x
x t




same different

21. CISE301_Topic8L1 21
Classification of ODEs
ODEs can be classified in different ways:
 Order
 First order ODE
 Second order ODE
 Nth order ODE
 Linearity
 Linear ODE
 Nonlinear ODE
 Auxiliary conditions
 Initial value problems
 Boundary value problems

22. CISE301_Topic8L1 22
Analytical Solutions
 Analytical Solutions to ODEs are available
for linear ODEs and special classes of
nonlinear differential equations.

23. CISE301_Topic8L1 23
Numerical Solutions
 Numerical methods are used to obtain a
graph or a table of the unknown function.
 Most of the Numerical methods used to
solve ODEs are based directly (or
indirectly) on the truncated Taylor series
expansion.

24. CISE301_Topic8L1 24
Classification of the Methods
Numerical Methods
for Solving ODE
Multiple-Step Methods
Estimates of the solution
at a particular step are
based on information on
more than one step
Single-Step Methods
Estimates of the solution
at a particular step are
entirely based on
information on the
previous step