2. Slide number 2
Ordinary Differential Equations
Where do ODEs arise?
Notation and Definitions
Solution methods for 1st order
ODEs
3. Slide number 3
Where do ODE’s arise
All branches of Engineering
Economics
Biology and Medicine
Chemistry, Physics etc
Anytime you wish to find out how
something changes with time (and
sometimes space)
4. Slide number 4
Example – Newton’s Law of
Cooling
This is a model of how the
temperature of an object changes as
it loses heat to the surrounding
atmosphere:
Temperature of the object: Obj
T Room Temperature: Room
T
Newton’s laws states: “The rate of change in the temperature of an
object is proportional to the difference in temperature between the object
and the room temperature”
)
( Room
Obj
Obj
T
T
dt
dT
Form
ODE
t
Room
init
Room
Obj e
T
T
T
T
)
(
Solve
ODE
Where is the initial temperature of the object.
init
T
5. Slide number 5
Notation and Definitions
Order
Linearity
Homogeneity
Initial Value/Boundary value
problems
6. Slide number 6
Order
The order of a differential
equation is just the order of
highest derivative used.
0
2
2
dt
dy
dt
y
d
.
2nd order
3
3
dt
x
d
x
dt
dx
3rd order
7. Slide number 7
Linearity
The important issue is how the
unknown y appears in the equation.
A linear equation involves the
dependent variable (y) and its
derivatives by themselves. There
must be no "unusual" nonlinear
functions of y or its derivatives.
A linear equation must have constant
coefficients, or coefficients which
depend on the independent variable
(t). If y or its derivatives appear in the
coefficient the equation is non-linear.
8. Slide number 8
Linearity - Examples
0
y
dt
dy
is linear
0
2
x
dt
dx
is non-linear
0
2
t
dt
dy
is linear
0
2
t
dt
dy
y is non-linear
9. Slide number 9
Linearity – Summary
y
2
dt
dy
dt
dy
y
dt
dy
t
2
dt
dy
y
t)
sin
3
2
( y
y )
3
2
( 2
Linear Non-linear
2
y )
sin( y
or
10. Slide number 10
Linearity – Special Property
If a linear homogeneous ODE has solutions:
)
(t
f
y )
(t
g
y
and
then:
)
(
)
( t
g
b
t
f
a
y
where a and b are constants,
is also a solution.
11. Slide number 11
Linearity – Special Property
Example:
0
2
2
y
dt
y
d
0
sin
sin
sin
)
(sin
2
2
t
t
t
dt
t
d
0
cos
cos
cos
)
(cos
2
2
t
t
t
dt
t
d
0
cos
sin
cos
sin
cos
sin
)
cos
(sin
2
2
t
t
t
t
t
t
dt
t
t
d
t
y sin
t
y cos
has solutions and
Check
t
t
y cos
sin
Therefore is also a solution:
Check
12. Slide number 12
Homogeniety
Put all the terms of the equation
which involve the dependent variable
on the LHS.
Homogeneous: If there is nothing
left on the RHS the equation is
homogeneous (unforced or free)
Nonhomogeneous: If there are
terms involving t (or constants) - but
not y - left on the RHS the equation
is nonhomogeneous (forced)
13. Slide number 13
Example
1st order
Linear
Nonhomogeneous
Initial value problem
g
dt
dv
0
)
0
( v
v
w
dx
M
d
2
2
0
)
(
0
)
0
(
and
l
M
M
2nd order
Linear
Nonhomogeneous
Boundary value
problem
14. Slide number 14
Example
2nd order
Nonlinear
Homogeneous
Initial value problem
2nd order
Linear
Homogeneous
Initial value problem
0
sin
2
2
2
dt
d
0
)
0
(
,
0 0
dt
d
θ
)
θ(
0
2
2
2
dt
d
0
)
0
(
,
0 0
dt
d
θ
)
θ(
15. Slide number 15
Solution Methods - Direct
Integration
This method works for equations
where the RHS does not depend on
the unknown:
The general form is:
)
(t
f
dt
dy
)
(
2
2
t
f
dt
y
d
)
(t
f
dt
y
d
n
n
16. Slide number 16
Direct Integration
y is called the unknown or
dependent variable;
t is called the independent variable;
“solving” means finding a formula for
y as a function of t;
Mostly we use t for time as the
independent variable but in some
cases we use x for distance.
17. Slide number 17
Direct Integration – Example
Find the velocity of a car that is
accelerating from rest at 3 ms-2:
3
a
dt
dv
c
t
v
3
t
v
c
c
v
3
0
0
3
0
0
)
0
(
If the car was initially at rest we
have the condition:
18. Slide number 18
Solution Methods - Separation
The separation method applies only to
1st order ODEs. It can be used if the
RHS can be factored into a function of t
multiplied by a function of y:
)
(
)
( y
h
t
g
dt
dy
19. Slide number 19
Separation – General Idea
First Separate:
dt
t
g
y
h
dy
)
(
)
(
Then integrate LHS with
respect to y, RHS with respect
to t.
C
dt
t
g
y
h
dy
)
(
)
(
20. Slide number 20
Separation - Example
)
sin( t
y
dt
dy
Separate:
dt
t
dy
y
)
sin(
1
Now integrate:
)
cos(
)
cos(
)
cos(
)
ln(
)
sin(
1
t
c
t
Ae
y
e
y
c
t
y
dt
t
dy
y
