and d, p, q are nonzero, then the differential equationMathematical Methods (10/24.539)
VIII. Special Functions
Equations Solvable in Terms of Bessel Functions
If and d, p, q are nonzero, then the differential equation
has complete solution
where
with conditions:
An exception to the above rule exists only when the equation reduces exactly to a second-order Euler-Cauchy equation of the form
which has solutions in the form y = xm (see
Section II of these notes).The expressions summarized in eqns. (8.59) - (8.61) represent a recipe for analytically solving a wide class of problems in terms of the ordinary or modified Bessel functions. Many 2nd order variable coefficient linear systems can be cast into this form and, if this can be done, the above equations represent a systematic approach for solving these systems. Two specific examples,
Example 8.1 and Example 8.2, illustrate the use of this general relationship.A third problem involving Bessel functions,
Example 8.3, is also available. This example illustrates the use of the modified Bessel functions to get an analytical solution to a simple heat conduction problem in cylindrical coordinates. Plotting of the resultant temperature and gradient profiles is done in a simple Matlab file as another example showing the evaluation of the Bessel functions within the Matlab environment.|
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10/24.539 Lecture Notes by Dr. J. R. White, UMass-Lowell (updated October 1998).
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