TY - JOUR

T1 - Generating Resonant and Repeated Root Solutions to Ordinary Differential Equations Using Perturbation Methods

AU - Gouveia, Bernardo

AU - Stone, Howard A.

N1 - Funding Information:
\ast Received by the editors February 1, 2021; accepted for publication (in revised form) June 21, 2021; published electronically May 5, 2022. https://doi.org/10.1137/21M1395922 Funding: The work of the first author was supported by a Paul and Daisy Soros Fellowship and the NSF Graduate Research Fellowship Program. \dagger Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08544 USA (bgouveia@princeton.edu). \ddagger Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544 USA (hastone@princeton.edu).
Publisher Copyright:
\bigcirc c 2022 Society for Industrial and Applied Mathematics

PY - 2022

Y1 - 2022

N2 - In the study of ordinary differential equations (ODEs) of the form L\^[y(x)] = f(x), where L\^ is a linear differential operator, two related phenomena can arise: resonance, where f(x) \propto u(x) and L\^[u(x)] = 0, and repeated roots, where f(x) = 0 and L\^ = D\^ n for n \geq 2. We illustrate a method to generate exact solutions to these problems by taking a known homogeneous solution u(x), introducing a parameter \epsilon such that u(x) \rightarrow u(x; \epsilon ), and Taylor expanding u(x; \epsilon ) about \epsilon = 0. The coefficients of this expansion \partial\partial \epsilonkku\bigm| \bigm|\epsilon=0 yield the desired resonant or repeated root solutions to the ODE. This approach, whenever it can be applied, is more insightful and less tedious than standard methods such as reduction of order or variation of parameters. We provide examples of many common ODEs, including constant coefficient, equidimensional, Airy, Bessel, Legendre, and Hermite equations. While the ideas can be introduced at the undergraduate level, we could not find any existing elementary or advanced text that illustrates these ideas with appropriate generality.

AB - In the study of ordinary differential equations (ODEs) of the form L\^[y(x)] = f(x), where L\^ is a linear differential operator, two related phenomena can arise: resonance, where f(x) \propto u(x) and L\^[u(x)] = 0, and repeated roots, where f(x) = 0 and L\^ = D\^ n for n \geq 2. We illustrate a method to generate exact solutions to these problems by taking a known homogeneous solution u(x), introducing a parameter \epsilon such that u(x) \rightarrow u(x; \epsilon ), and Taylor expanding u(x; \epsilon ) about \epsilon = 0. The coefficients of this expansion \partial\partial \epsilonkku\bigm| \bigm|\epsilon=0 yield the desired resonant or repeated root solutions to the ODE. This approach, whenever it can be applied, is more insightful and less tedious than standard methods such as reduction of order or variation of parameters. We provide examples of many common ODEs, including constant coefficient, equidimensional, Airy, Bessel, Legendre, and Hermite equations. While the ideas can be introduced at the undergraduate level, we could not find any existing elementary or advanced text that illustrates these ideas with appropriate generality.

KW - ordinary differential equations

KW - repeated roots

KW - resonance

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U2 - 10.1137/21M1395922

DO - 10.1137/21M1395922

M3 - Article

AN - SCOPUS:85130620872

VL - 64

SP - 485

EP - 499

JO - SIAM Review

JF - SIAM Review

SN - 0036-1445

IS - 2

ER -