Industrial Engineering Class Notes 25 Power Series Solutions Part 7 Earlier The

Differential Equations

for Engineers:

the Essentials

Class 25 notes

Agenda: Class 25

Review Homework Assignment 23

Lectures:

Partial Differential Equations:

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:

It suffices to solve Equation 1 with the initial condition . We will

have more to say about the initial value of the derivative momentarily.

1)0( =R

Power Series Solutions: Part I (2)

We solve Equation 1 in terms of an infinite series:

Using Equation 2

n

nxcxR 



=

0

)(

Equation 2

=

2

n

Power Series Solutions: Part I (3)

Using Equations 2, 4 and 5 in Equation 1, we have

For this equation to be true for all the coefficients of each power must

be zero. From this fact we can immediately see that

x

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

single sum:

To do this we must replace with in the first sum in Eq’n 9

and with in the second. The result is

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

Power Series Solutions: Part I (6)

In general the solution to Equation 1 can be described as

or as

Power Series Solutions: Part I (7)

Earlier in the course we learned that the equation

with specified initial conditions and

has a solution of the form

with and given by the solution to

00 )( yty =

00 )( yt

dt

dy 

=

c

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

is called a Bessel function of the second kind of order zero. It can be

written in the form

Partial Differential Equations

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

It is a 2nd order linear ODE and so has, of course, two fundamental

solutions. The solution having finite and at the origin is

the Bessel function of the first kind of order m, described by the series

y

dxdy /

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

origin. It is needed, for example, in problems where boundary

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

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