Riccati Equation

a first-order ordinary differential equation of the form

where a, b, and a are constants. This equation was first investigated by J. Riccati in 1724; certain special cases were studied earlier. In work done in 1724 and 1725, D. Bernoulli established that equation (*) can be integrated in terms of elementary functions when α = – 2 or α = – 4k/(2k – 1), where k is an integer. J. Liouville proved in 1841 that, for other values of α, the equation (*) cannot be solved by quadrature, that is, by applying a finite number of algebraic operations, transformations of variables, and indefinite integrations to elementary functions; a general solution can be expressed in terms of cylindrical functions.

The differential equation

where P(x), Q(x), and R(x) are continuous functions, is called the generalized Riccati equation. When P(x) = 0, the generalized Riccati equation is a linear differential equation; when R(x) = 0, it is the Bernoulli equation. In these two cases, the equation is integrable in closed form. Other cases of the integrability of the generalized Riccati equation have also been studied.