Differential Equations
for Engineers:
the Essentials
Class 25 notes
Agenda: Class 25
Review Homework Assignment 23
Lectures:
Partial Differential Equations:
Power Series Solutions –Part 1
Bessel Functions
Partial Differential Equations
Power Series Solutions
Part I
Power Series Solutions: Part I
Our objective here is to show how to solve problems exemplified by
Bessel’s equation of order zero:
0
1
2
2
=++ R
dx
dR
xdx
Rd
Equation 1
Power Series Solutions: Part I (2)
We solve Equation 1 in terms of an infinite series:
The initial condition implies that
Using Equation 2
n
n
nxcxR
=
=
0
)(
Equation 2
−
=
−
=
==
2
1
1
1
11
n
n
n
n
n
n
xncxnc
xdx
dR
x
Equation 4
1)0( =R
Power Series Solutions: Part I (3)
Using Equations 2, 4 and 5 in Equation 1, we have
=
=
−
=
−
=++−
01
2
2
2
0)1(
n
n
n
n
n
n
n
n
n
xcxncxcnn
For this equation to be true for all the coefficients of each power must
be zero. From this fact we can immediately see that
Equations 2 and 7 together imply that
x
0
1=c
Eq’n 7
Power Series Solutions: Part I (4)
We are left with
The next step is an important one in solving this problem –we must
manipulate the indices so that the equation can be written as a
To do this we must replace with in the first sum in Eq’n 9
and with in the second. The result is
=
=
−=+
02
22 0
n
n
n
n
n
nxcxcn
Eq’n 9
2−n
m
n
m
Power Series Solutions: Part I (5)
Again, for Equation 11 to be true for all the coefficients of each
power must be zero. This implies the recursion relation
Using Equations 3, 7 and 12, we find that for odd mand
x
0=
m
c
22
0
2
2
2
1
)1(
2
1
2
1
−=−=−=
cc
Power Series Solutions: Part I (6)
In general the solution to Equation 1 can be described as
or as
n
n
nx
n
xR
2
0
22)!(
)1(
)(
−
=
=
Eq’n 13
Power Series Solutions: Part I (7)
Earlier in the course we learned that the equation
with and given by the solution to
0)()(
2
2
=++ ytq
dt
dy
tp
dt
yd
1
c
2
c
)()(
ytyctyc
=+
Power Series Solutions: Part I (8)
The Bessel equation does not meet this proviso at the origin.
It is not possible to solve the Bessel equation with arbitrary initial
conditions as in Equation 15 when the solution domain includes
the origin.
Power Series Solutions: Part I (9)
The second fundamental solution to Bessel’s equation of order zero
See the Supplement to Class 25 Notes for proof of this.
0
1
2
2
=++ R
dx
dR
xdx
Rd
Partial Differential Equations
Bessel Functions
Bessel Functions
Bessel’s Equations are ordinary differential equations that derive from
solving the wave equation by separation of variables in cylindrical
coordinates. Bessel’s Equation of order mis
We derived the form for the zero order case in the last class. Later in this
class we will derive the formula for order m.
0)/1(
122
2
2
=−++ yxm
dx
dy
xdx
yd
Equation 1
Bessel Functions of the First Kind:
Orders 0, 1 and 2
Bessel Functions: Second Solution
The other fundamental solution, called the Bessel function of the
second kind of order m, has infinite and at the
The second solution to Equation 1 can be written
y
dxdy /
Bessel Function of First Kind of Order Zero
The Bessel functions of the first kind are easy to compute. For
example, for small , can be expressed as
For large
…
26
1
22
1
2
1)(
6
2
4
2
2
0+
−
+
−= xxx
xJ
x
x
)(
0xJ
Bessel Functions (10)
Mathematicians traditionally use a slightly different form of the second
solution that is called that has the advantage of a
complementary relationship to for large :
)( xYm
x
)( xJm