College of Engineering and Computer Science
Mechanical Engineering Department
Mechanical Engineering 501A
Seminar in Engineering Analysis
Fall 2004 Number: 17472 Instructor: Larry Caretto
Goals and Assignments for the Eighth Week
Eighth week goals
We know that the solution to the differential equation d 2/dx2 + 2y = 0 is A sin x + B cos x.
However we also know that we can express the sine and cosine as an infinite series. Thus we
could write the solution to d2/dx2 + 2y = 0 as an infinite series. We want to see how we can
generalize this idea and develop an infinite-series solution to a differential equation when we do
not have an analytical solution. To explore this idea further, we will examine the solutions to
other differential equations with engineering applications, such as Bessel’s equation. First we
have to develop a method, Frobenius method of power series solutions to solve this equation.
After studying this week’s material you should be able to obtain a power series solution for a
differential equation including the use of the Frobenius method.
Bessel’s equation, x2d2y/dx2 + xdy/dx + (x2 – 2)y = 0 has a power series solution that occurs
frequently in engineering problems that its series solutions are tabulated functions. We will obtain
the series solution that defines the tabulated Bessel functions, J(x) and Y(x). You should be
able to recognize Bessel’s equation, and write its solutions in terms of the tabulated functions as
y = A J(x) + B Y(x). You should also be able to apply formulae for derivatives and integrals of
Bessel functions and transform other differential equations into Bessel’s equation.
Reading for October 12, 2004
Pages 194-203 review the basic ideas of power series, including convergence of partial sums and
various operations on power series.
Pages 203-208 show the basic technique of power series solutions that can be applied when the
coefficients of the power series are analytic. (That is the coefficients can be expressed as a
power series.) This solution technique is applied to Legendre’s equation, which arises in
engineering problems written in a spherical coordinate system.
Pages 211-217 describe the Frobenius method for power series solution of differential equations
that can applied in limited circumstances when coefficients in the differential equation are not
analytic.
Reading for October 14, 2004
Pages 218-232 use the Frobenius method to solve Bessel’s equation. This presents the Bessel
function as a set of functions that arises from a differential equation solution. This function mainly
arises in engineering applications expressed in cylindrical coordinate systems.
Homework due October 19, 2004
Page 204, problem 5. This problem illustrates the basic approach to obtaining a power series
solution to a differential equation.
Page 205, problem 23. This problem gives you he exercise of shifting the index in a summation;
this is an important step in doing power series solutions.
Engineering Building Room 1333
Email: lcaretto@csun.edu
Mail Code
8348
Phone: 818.677.6448
Fax: 818.677.7062
Assignments for week eight
ME 501A, L. S. Caretto, Fall 2004
Page 2
Page 216, problem 9. This is an application of the Frobenius method to the solution of a
differential equation.
Page 226, problem 23. This application of integral formulas for Bessel functions is important in
solutions of differential equations involving Bessel function expansions.
Page 232, problems 7 and 14. These problems show how some differential equations can be
transformed into equations that can be solved in terms of Bessel functions. Problem 14
introduces the modified Bessel function that arises in the analysis of heat transfer from circular
fins.