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On the Formulaic Solution of a \((n+1)\)th Order Differential Equation

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Abstract

A systematic method to facilitate the solution of a \((n+1)\)th order differential equation is introduced. This set of ordinary differential equations is unique in the sense that it defines a hierarchy whose solution can be manipulated to solve a partial differential hierarchy, called Burgers’ hierarchy. The latter forms a highly nonlinear evolutionary class of equations. Our approach can be described as in the following two ways. Firstly, we construct the one-parameter Lie group of transformations that leave the ordinary differential equation invariant, and find that they are universally derived from the \((n+1)\) complex roots of a certain polynomial. Secondly, we prove that the exact solution of the hierarchy for all values of n, is given by an integration formula. An important consequence of this study, is the extension of the formulaic method to exactly solve the entire Burgers’ hierarchy. We exemplify the advantageous nature of our results through solving various members of the hierarchy.

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  1. School of Mathematics, University of the Witwatersrand, Johannesburg, 2001, South Africa

    U. Obaidullah & Sameerah Jamal

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  1. U. Obaidullah
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  2. Sameerah Jamal
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Correspondence to Sameerah Jamal.

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SJ acknowledges the financial support of the National Research Foundation of South Africa (118047).

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Obaidullah, U., Jamal, S. On the Formulaic Solution of a \((n+1)\)th Order Differential Equation. Int. J. Appl. Comput. Math 7, 58 (2021). https://doi.org/10.1007/s40819-021-01010-9

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  • DOI: https://doi.org/10.1007/s40819-021-01010-9

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