Remarks on the uniqueness of second order ODEs

Dalibor Pražák

Applications of Mathematics (2011)

  • Volume: 56, Issue: 1, page 161-172
  • ISSN: 0862-7940

Abstract

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We are concerned with the uniqueness problem for solutions to the second order ODE of the form x ' ' + f ( x , t ) = 0 , subject to appropriate initial conditions, under the sole assumption that f is non-decreasing with respect to x , for each t fixed. We show that there is non-uniqueness in general; on the other hand, several types of reasonable additional assumptions make the problem uniquely solvable. The interest in this problem comes, among other, from the study of oscillations of lumped parameter systems with implicit constitutive relations.

How to cite

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Pražák, Dalibor. "Remarks on the uniqueness of second order $\rm ODEs$." Applications of Mathematics 56.1 (2011): 161-172. <http://eudml.org/doc/116509>.

@article{Pražák2011,
abstract = {We are concerned with the uniqueness problem for solutions to the second order ODE of the form $x^\{\prime \prime \}+f(x,t)=0$, subject to appropriate initial conditions, under the sole assumption that $f$ is non-decreasing with respect to $x$, for each $t$ fixed. We show that there is non-uniqueness in general; on the other hand, several types of reasonable additional assumptions make the problem uniquely solvable. The interest in this problem comes, among other, from the study of oscillations of lumped parameter systems with implicit constitutive relations.},
author = {Pražák, Dalibor},
journal = {Applications of Mathematics},
keywords = {second order ODEs; uniqueness of solutions; oscillations; second order ODEs; uniqueness of solutions; oscillation},
language = {eng},
number = {1},
pages = {161-172},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Remarks on the uniqueness of second order $\rm ODEs$},
url = {http://eudml.org/doc/116509},
volume = {56},
year = {2011},
}

TY - JOUR
AU - Pražák, Dalibor
TI - Remarks on the uniqueness of second order $\rm ODEs$
JO - Applications of Mathematics
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 1
SP - 161
EP - 172
AB - We are concerned with the uniqueness problem for solutions to the second order ODE of the form $x^{\prime \prime }+f(x,t)=0$, subject to appropriate initial conditions, under the sole assumption that $f$ is non-decreasing with respect to $x$, for each $t$ fixed. We show that there is non-uniqueness in general; on the other hand, several types of reasonable additional assumptions make the problem uniquely solvable. The interest in this problem comes, among other, from the study of oscillations of lumped parameter systems with implicit constitutive relations.
LA - eng
KW - second order ODEs; uniqueness of solutions; oscillations; second order ODEs; uniqueness of solutions; oscillation
UR - http://eudml.org/doc/116509
ER -

References

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  1. Hartman, P., Ordinary Differential Equations. 2nd ed. with some corrections and additions, S. M. Hartman Baltimore (1973). (1973) Zbl0281.34001MR0344555
  2. Meirovitch, L., Elements of Vibration Analysis. Second edition, McGraw-Hill New York (1986). (1986) 
  3. Pražák, D., Rajagopal, K. R., Mechanical oscillators described by a system of differential-algebraic equations, Submitted. 
  4. Rajagopal, K. R., A generalized framework for studying the vibration of lumped parameter systems, Submitted. 

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