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MATH 2120
MATHEMATICAL METHODS FOR
DIFFERENTIAL EQUATIONS
LECTURE NOTES
2
Introduction and Review
Ordinary Differential Equations (ODEs) are equations that contain derivatives of
an unknown function with respect to an independent variable.
These often
describe some physical situation and their solution gives some insight into the
problem.
The principle task in differential equations and their applications is to find all
solutions of given equations and investigate their properties.
Notation: Ordinary Derivatives
)
x
(
y
dx
y
d
y
D
)
x
(
y
)
x
(
y
dx
y
d
y
D
)
x
(
y
)
x
(
y
dx
dy
Dy
)
n
(
n
n
n
)
2
(
2
2
2
)
1
(
!
!
!
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Definition: Order
An ODE is the order of the highest derivative that appears in the equation.
A general n
th
order ODE has the form
F(x, y, y
!
, y
"
, …, y
(n)
) = 0
Examples
g
dt
y
d
2
2
2
nd
order
p
dt
dp
#
1
st
order
4
Definition: Differential Operator
If we rearrange the ODE so that all the terms involving the unknown function
y(x) are on the left and all the terms not involving y(x) are on the right then
L
{y} = f(x)
where
L
is the differential operator.
Definition: Linear ODE
The differential operator
L
is a linear operator if
L
{c
1
y
1
(x) + c
2
y
2
(x)} =
L
{c
1
y
1
(x)} +
L
{c
2
y
2
(x)}
for all functions y
1
(x) and y
2
(x) and all constants c
1
and c
2
.
In general a linear ODE can be written in the form
$%
$% $%
$% $%$% $%
x
f
x
y
x
a
x
y
x
a
x
y
x
a
0
1
n
1
n
n
n
&
&
&
'
'
!
.
5
Definition: Homogeneous Linear ODE
A linear ODE of the form
L
{y(x)} = 0
Definition: Nonhomogeneous Linear ODE
A linear ODE of the form
L
{y(x)} = f(x) where f(x) ± 0
x
±
.
f(x) is called a forcing term.
Definition: Nonlinear ODE
An ODE of the form
L
{y(x)} = f(x) where
L
is
not
a linear operator.
Examples
0
x
sin
dx
dy
x
dx
y
d
y
2
2
&
&
2
nd
order nonlinear ODE.
0
x
sin
dx
dy
x
dx
y
d
2
2
&
&
2
nd
order linear nonhomogeneous ODE.
6
Initial Value and Boundary Value Problems
In most applications you need a particular solution rather that a general one.
When a function is integrated, an arbitrary integration constant is introduced.
Thus, when you integrate a
first order
ODE, the general solution will contain
one unknown arbitrary constant
, which could be specified using a single
condition to obtain the particular solution.
For
second order
ODES we have
two arbitrary constants
, and so we require
two conditions to specify the unknowns.
Often these conditions come in the form of initial values, where both the value of
the unknown function and its derivative are given for a given value of the
variable, e.g.
1
0
0
0
K
)
x
(
y
,
K
)
x
(
y
!
where x = x
0
is a given point and K
0
and K
1
are given numbers.
These are called
initial conditions
.
These, together with the ODE form an
initial value problem
.
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Some applications lead to conditions of the type
y(P
1
) = k
1
,
y(
P
2
) = k
2
.
These are known as
boundary conditions
, since they refer to the endpoints P
1
and P
2
(or boundary points P
1
and P
2
) of an interval I on which the ODE is
considered.
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This note was uploaded on 09/03/2011 for the course MATH 2120 taught by Professor Dr.trevorlanglands during the One '09 term at University of New South Wales.
 One '09
 Dr.TrevorLanglands
 Differential Equations, Equations, Derivative

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