Riccati equation – Wikipedia
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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.
Conversion to a second order linear equation[edit]
The non-linear Riccati equation can always be converted 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
since
so that
and hence
A solution of this equation will lead to a solution
of the original Riccati equation.
Application to the Schwarzian equation[edit]
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 quadrature[edit]
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
Substituting
in the Riccati equation yields
and since
it follows that
or
which is a Bernoulli equation. The substitution that is needed to solve this Bernoulli equation is
Substituting
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 also[edit]
References[edit]
Further reading[edit]
- 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 links[edit]
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