Chapter 6
Series solutions of ODEs and special
functions
Syllabus section:
5. Series solution of ODEs. Introduction to special functions, e.g., Legendre, Bessel, and Hermite functions;
orthogonality of special functions.
6.1
Context
[This section is not in itself examinable, although it may be necessary to understand what follows.]
This chapter is firstly about some methods for obtaining series solutions of ordinary differential equations,
a particular application of which will be in solving Laplace’s equation later in the course. These methods also
relate to socalled ‘special functions’, which are discussed in the second half of this chapter, and very fully in
major texts
1
and which, in the days before computers, were even more important than they are now, because
they provided a way to solve problems we could now tackle using numerical computer programs.
To understand where this fits in, we need to know some basic things about differential equations. That
is what this section covers, in particular for the benefit of any who have not done the firstyear differential
equations course.
A (scalar)
differential equation
(DE) is some equation in a function,
u
say, and its derivatives, which we
want to solve for
u
.
This is a scalar equation because
u
is a scalar function. One can also have DEs for vectors or matrices,
which are equivalent to systems of DEs for the components, but we will not discuss these here.
It is an
ordinary differential equation
(ODE) if
u
depends only on a single variable, say
x
. In an ODE
we often write
u
′
for d
u
/
d
x
, and
u
(
k
)
≡
d
k
u
/
d
x
k
. If
u
depends on more than one variable, the derivatives are
partial derivatives and the equation is a
partial differential equation
(PDE).
The
order
of a differential equation is the order of the highest derivative that it contains, and the
degree
1
E.g. “Handbook of Mathematical Functions” by M. Abramowitz and I. Stegun, which has over 1000 pages, including extensive
tables of numerical values.
61
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View Full Documentis the power to which that derivative appears when fractional powers have been eliminated. An equation is
called linear if any term in it has at most one of
u
,
u
′
, ...,
u
(
n
)
, to degree one at most.
Equation
Type
Order
Degree
Linear?
d
u
d
x
+
p
(
x
)
u
+
q
(
x
)
u
2
=
r
(
x
)
ODE
1
1
No
p
(
x
)
,
q
(
x
)
and
r
(
x
)
known functions. Riccati equation
x
2
d
2
u
d
x
2
+
x
d
u
d
x
+ (
x
2
−
n
2
)
u
=
0
ODE
2
1
Yes
n
a constant. Bessel’s equation.
d
2
u
d
x
2
b
1
+
p
d
u
d
x
P
2
B
3
/
2
=
g
(
x
)
ODE
2
2
No
g
(
x
)
some known function. Equation for curvature of a curve.
∂
2
u
x
2
=
κ
u
t
PDE
2
1
constant,
u
=
u
(
x
,
t
)
. Heat equation or diffusion equation.
Table 6.1: Some examples of differential equations
We shall assume for simplicity that we always can write our ODEs in the form
d
n
u
d
x
n
=
w
(
x
,
u
,
u
′
u
(
n
−
1
)
)
(6.1)
where
w
is some known function of its arguments, and that we always do do this.
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 Spring '09
 Magnetism, ORDINARY DIFFERENTIAL EQUATIONS, Partial differential equation, Regular singular point

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