TITLE:
A General Analytical Solution to the Nonlinear Riccati Differential Equation
AUTHORS:
Leandro Meléndez-Lugo
KEYWORDS:
Differential Equations, Serendipity, Solution Methods, Riccati Equation, Differential Calculus
JOURNAL NAME:
Journal of High Energy Physics, Gravitation and Cosmology,
Vol.11 No.4,
September
26,
2025
ABSTRACT: The nonlinear Riccati differential equation, since its enunciation by Jacopo Riccati in 1724, has become a legendary equation. Despite a 300-year search for a solution, it has not been possible to solve it analytically and completely. Brilliant mathematicians from various illustrious eras of differential equations have been unsuccessful. This work describes an effective method for obtaining a solution to the general nonlinear Riccati equation. The method is divided into two fundamental parts. The first part consists of transforming the general Riccati equation into a simplified expression known as the canonical Riccati equation. The second part, much more strategic and involving a kind of serendipity, also involves transforming the Riccati equation. That is, it entails the creation of an additional nonlinear differential equation, also of the first order, that can be solved relatively easily. Its solutions allow us to obtain the solution to the canonical Riccati equation, and ultimately, the solution to the general equation.