Riccati equation

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In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form

where and . If the equation reduces to a Bernoulli equation, while if the equation becomes a first order linear ordinary differential equation.

The equation is named after Jacopo Riccati (1676–1754).[1]

More generally, the term Riccati equation is used to refer to matrix equations with an analogous quadratic term, which occur in both continuous-time and discrete-time linear-quadratic-Gaussian control. The steady-state (non-dynamic) version of these is referred to as the algebraic Riccati equation.

Reduction to a second order linear equationEdit

The non-linear Riccati equation can always be reduced to a second order linear ordinary differential equation (ODE):[2] If


then, wherever   is non-zero and differentiable,   satisfies a Riccati equation of the form


where   and  , because


Substituting  , it follows that   satisfies the linear 2nd order ODE




so that


and hence


A solution of this equation will lead to a solution   of the original Riccati equation.

Application to the Schwarzian equationEdit

An important application of the Riccati equation is to the 3rd order Schwarzian differential equation


which occurs in the theory of conformal mapping and univalent functions. In this case the ODEs are in the complex domain and differentiation is with respect to a complex variable. (The Schwarzian derivative   has the remarkable property that it is invariant under Möbius transformations, i.e.   whenever   is non-zero.) The function   satisfies the Riccati equation


By the above   where   is a solution of the linear ODE


Since  , integration gives   for some constant  . On the other hand any other independent solution   of the linear ODE has constant non-zero Wronskian   which can be taken to be   after scaling. Thus


so that the Schwarzian equation has solution  

Obtaining solutions by quadratureEdit

The correspondence between Riccati equations and second-order linear ODEs has other consequences. For example, if one solution of a 2nd order ODE is known, then it is known that another solution can be obtained by quadrature, i.e., a simple integration. The same holds true for the Riccati equation. In fact, if one particular solution   can be found, the general solution is obtained as




in the Riccati equation yields


and since


it follows that




which is a Bernoulli equation. The substitution that is needed to solve this Bernoulli equation is




directly into the Riccati equation yields the linear equation


A set of solutions to the Riccati equation is then given by


where z is the general solution to the aforementioned linear equation.

See alsoEdit


  1. ^ Riccati, Jacopo (1724) "Animadversiones in aequationes differentiales secundi gradus" (Observations regarding differential equations of the second order), Actorum Eruditorum, quae Lipsiae publicantur, Supplementa, 8 : 66-73. Translation of the original Latin into English by Ian Bruce.
  2. ^ Ince, E. L. (1956) [1926], Ordinary Differential Equations, New York: Dover Publications, pp. 23–25

Further readingEdit

  • Hille, Einar (1997) [1976], Ordinary Differential Equations in the Complex Domain, New York: Dover Publications, ISBN 0-486-69620-0
  • Nehari, Zeev (1975) [1952], Conformal Mapping, New York: Dover Publications, ISBN 0-486-61137-X
  • Polyanin, Andrei D.; Zaitsev, Valentin F. (2003), Handbook of Exact Solutions for Ordinary Differential Equations (2nd ed.), Boca Raton, Fla.: Chapman & Hall/CRC, ISBN 1-58488-297-2
  • Zelikin, Mikhail I. (2000), Homogeneous Spaces and the Riccati Equation in the Calculus of Variations, Berlin: Springer-Verlag
  • Reid, William T. (1972), Riccati Differential Equations, London: Academic Press

External linksEdit