Electron. J. Diff. Equ., Vol. 2015 (2015), No. 175, pp. 1-15.

Oscillatory solutions of the Cauchy problem for linear differential equations

Gro Hovhannisyan, Oliver Ruff

Abstract:
We consider the Cauchy problem for second and third order linear differential equations with constant complex coefficients. We describe necessary and sufficient conditions on the data for the existence of oscillatory solutions. It is known that in the case of real coefficients the oscillatory behavior of solutions does not depend on initial values, but we show that this is no longer true in the complex case: hence in practice it is possible to control oscillatory behavior by varying the initial conditions. Our Proofs are based on asymptotic analysis of the zeros of solutions, represented as linear combinations of exponential functions.

Submitted May 6, 2015. Published June 24, 2015.
Math Subject Classifications: 34C10.
Key Words: Linear ordinary differential equation; oscillation; initial value problem; characteristic polynomial; characteristic roots.

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Gro Hovhannisyan
Kent State University at Stark
6000 Frank Ave. NW
Canton, OH 44720-7599, USA
email: ghovhann@kent.edu
  Oliver Ruff
Kent State University at Stark
6000 Frank Ave. NW
Canton, OH 44720-7599, USA
email: oruff@kent.edu

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