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Oscillation of an Euler-Cauchy dynamic equation

Pages: 423 - 431, Issue Special, July 2003

 Abstract        Full Text (169.8K)              

S. Huff - University of Nebraska-Lincoln, Lincoln, NE 65888-0323, United States (email)
G. Olumolode - University of Nebraska-Lincoln, Lincoln, NE 65888-0323, United States (email)
N. Pennington - University of Nebraska-Lincoln, Lincoln, NE 65888-0323, United States (email)
A. Peterson - University of Nebraska-Lincoln, Lincoln, NE 65888-0323, United States (email)

Abstract: The Euler-Cauchy differential equation and difference equation are well known. Here we study a more general Euler-Cauchy dynamic equation. For this more general equation when we have complex roots of the corresponding characteristic equation we for the first time write solutions of this dynamic equation in terms of a generalized exponential function and generalized sine and cosine functions. This result is even new in the difference equation case. We then spend most of our time studying the oscillation properties of the Euler-Cauchy dynamic equation. Several oscillation results are given and an open problem is posed.

Keywords:  Euler¨CCauchy dynamic equation, time scale, oscillation.
Mathematics Subject Classification:  Primary: 39A10; Secondary: 34B10.

Received: September 2002; Published: April 2003.