Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 21 (2026), 1 -- 8
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This work is licensed under a Creative Commons Attribution 4.0 International License.

EXTENDED GENERALIZATION OF THE BERNOULLI EQUATION

Alexandre M. Afonso

Abstract. This short note introduce an analytic solution for a generalized Bernoulli non-linear first-order ordinary differential equation (ODE), extending the framework proposed by Azevedo and Valentino [Int J Math Educ Sci, 2016, 47(8), 1271--1276]. The family of analytic solutions are obtained by employing a non-linear substitution that transform the extended Bernoulli ODE into a classical Bernoulli equation. We derive a family of solutions and provide illustrative examples, demonstrating the versatility of our approach for arbitrary values of the Bernoulli non-linear term power α. This work connects to applications in population growth models, such as the θ-logistic and Richards models.

2020 Mathematics Subject Classification: 34A05; 34A06; 34A34
Keywords: Extended Bernoulli equation; differential equations; non-linear ODEs; Richards models

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Alexandre M. Afonso, ORCID: 0000-0003-2825-0709
CEFT - Transport Phenomena Research Center, Department of Mechanical Engineering,
Faculdade de Engenharia, Universidade do Porto,
Rua Dr. Roberto Frias s/n, 4200-465 Porto, Portugal.
e-mail: aafonso@fe.up.pt




Received: February 2, 2026; Accepted: March 10, 2026;
Published electronically: March 13, 2026.