Mathematical Methods (10/24.539)
VIII. Special Functions
Example 8.1 -- Solution Using Elementary and Bessel Function Methods
Problem Description:
Find the general solution to the following equation using an elementary approach (linear constant coefficient system) and by using the general form of Bessel's equation:
Problem Solution:
Method 1 Elementary Solution
This is a constant coefficient linear 2nd order ODE. Therefore, we let 
, and the characteristic equation becomes 
with roots 
. Thus the general solution can be written as
Method 2 Bessel Function Solution
Comparing the defining ODE with the most general form of Bessel’s equation [see eqns. (8.59) - (8.61)], we have
By equating the coefficient of the y’(x) term (i.e. 
), we have a = b = 0. Therefore, the coefficient of the y(x) term becomes
but with a = b = 0, we have
Therefore, letting c = 0, d = 1, and q = 1 gives the desired equality.
With all the coefficients known and all the proper conditions satisfied, we can evaluate the constants in the general solution as follows:
Therefore, the solution to the original ODE becomes
This implies that the half-order Bessel functions must be related to the sine and cosine functions, since the solutions using the two different methods must be identical. From Problem 10.4 in the Schaum’s Outline Series on Advanced Mathematics, we have
Therefore the solution for y(x) can be written as
and this solution is of the form that we expect for a linear constant coefficient system. Although using Bessel functions is not the most efficient way to go for this problem, this example does illustrate that the Bessel functions are applicable to a wide variety of systems (they were originally identified as a set of solutions for variable coefficient systems and here they were used to solve a constant coefficient system).
|
|
|
|
|
|
10/24.539 Lecture Notes by Dr. J. R. White, UMass-Lowell (updated October 1998).
