General solution of the Riccati equation remains unsolved

Determine a general analytic solution for the Riccati differential equation y'(x) = f2(x) y(x)^2 + f1(x) y(x) + f0(x) for arbitrary coefficient functions f2(x), f1(x), and f0(x), beyond the special cases currently known to be solvable.

Background

The Riccati equation y'(x) = f2(x) y(x)2 + f1(x) y(x) + f0(x) is a fundamental nonlinear ordinary differential equation with broad applications across control theory, fluid dynamics, and quantum mechanics. Despite its historical significance and longstanding study, obtaining general solutions for arbitrary coefficient functions has been a persistent challenge.

The paper emphasizes that, as of the cited literature, only certain special cases are known to be solvable exactly. The work proposes an integrability condition and provides analytic solutions under this condition, but the introduction explicitly notes that the general problem remains unsolved outside special cases, framing a broader unresolved question in the field.

References

However, despite more than 300 years having elapsed since Count Jacopo Francesco Riccati first introduced the equation, the problem remains unsolved except in certain special cases [4, 5].

— Analytic General Solutions of the Riccati equation  (2510.19297 - Ji-Xiang, 22 Oct 2025) in Section 1. Introduction (page 1)

In fact, a recent survey identifies the general solvability of Riccati-type equations for the explicit implementation of feedback-type optimal solutions in games with state- and control-dependent noise as an open research direction.

— Policy Iteration for Linear-Quadratic Stochastic Differential Games with State- and Control-Dependent Noise  (2608.17940 - Handwerker et al., 18 Aug 2026) in Section 1, Introduction