# Riccati Equation

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## Riccati equation

[ri′käd·ē i‚kwā·zhən]*y*′ =

*A*

_{0}(

*x*) +

*A*

_{1}(

*x*)

*y*+

*A*

_{2}(

*x*)

*y*

^{2}; every second-order linear differential equation can be transformed into an equation of this form.

*dP*(

*t*)/

*dt*+

*P*(

*t*)

*F*(

*t*) +

*F*

^{ T }(

*t*)

*P*(

*t*) -

*P*(

*t*)

*G*(

*t*)

*R*

^{-1}(

*t*)

*G*

^{ T }(

*t*)

*P*(

*t*) +

*Q*(

*t*) = 0, which frequently arises in control and estimation theory.

## 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 = – 4*k*/(2*k* – 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.

### REFERENCE

Kamke, E.*Spravochnik po obyknovennym differentsial’nym uravneniiam*, 4th ed. Moscow, 1971. (Translated from German.)