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
SN - 0036-1445
VL - 64
SP - 485
EP - 499
JO - SIAM Review
JF - SIAM Review
IS - 2
ER -