Riemann's differential equation typically refers to a type of linear partial differential equation associated with Riemann surfaces and complex analysis. However, there isn't a single, universally recognized differential equation directly defined as "Riemann's differential equation." One prominent equation related to Riemann surfaces is the Riemann-Hilbert problem, which is a type of boundary value problem for holomorphic functions, involving a piecewise constant function defined on contours in the complex plane.
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Riemann's differential equation is a second-order ordinary differential equation with three regular singular points, normalized to zero, one and infinity by a Möbius transformation. Its local characteristic exponents are encoded by the Papperitz symbol and constrained by the Fuchs relation.