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 paper, 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.