A Riccati equation
is a first-order equation of the form
This equation was used by Count Riccati of Venice (1676-1754) to help in solving second-order ordinary differential equations.
Solving Riccati equations is considerably more difficult than solving linear ODEs.
Any Riccati equation can be transformed to a second-order linear ODE. If the latter can be solved explicitly, then a solution for the Riccati equation can be derived.