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
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.
^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.
^Ince, E. L. (1956) [1926], Ordinary Differential Equations, New York: Dover Publications, pp. 23–25
Further reading
Hille, Einar (1997) [1976], Ordinary Differential Equations in the Complex Domain, New York: Dover Publications, ISBN0-486-69620-0
MATLAB function for solving continuous-time algebraic Riccati equation.
SciPy has functions for solving the continuous algebraic Riccati equation and the discrete algebraic Riccati equation.
February 06, 2023
riccati, equation, mathematics, narrowest, sense, first, order, ordinary, differential, equation, that, quadratic, unknown, function, other, words, equation, form, displaystyle, where, displaystyle, displaystyle, displaystyle, equation, reduces, bernoulli, equ. 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 y x q 0 x q 1 x y x q 2 x y 2 x displaystyle y x q 0 x q 1 x y x q 2 x y 2 x where q 0 x 0 displaystyle q 0 x neq 0 and q 2 x 0 displaystyle q 2 x neq 0 If q 0 x 0 displaystyle q 0 x 0 the equation reduces to a Bernoulli equation while if q 2 x 0 displaystyle q 2 x 0 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 Contents 1 Conversion to a second order linear equation 2 Application to the Schwarzian equation 3 Obtaining solutions by quadrature 4 See also 5 References 6 Further reading 7 External linksConversion to a second order linear equation EditThe non linear Riccati equation can always be converted to a second order linear ordinary differential equation ODE 2 If y q 0 x q 1 x y q 2 x y 2 displaystyle y q 0 x q 1 x y q 2 x y 2 then wherever q 2 displaystyle q 2 is non zero and differentiable v y q 2 displaystyle v yq 2 satisfies a Riccati equation of the form v v 2 R x v S x displaystyle v v 2 R x v S x where S q 2 q 0 displaystyle S q 2 q 0 and R q 1 q 2 q 2 displaystyle R q 1 frac q 2 q 2 because v y q 2 y q 2 y q 2 q 0 q 1 y q 2 y 2 q 2 v q 2 q 2 q 0 q 2 q 1 q 2 q 2 v v 2 displaystyle v yq 2 y q 2 yq 2 q 0 q 1 y q 2 y 2 q 2 v frac q 2 q 2 q 0 q 2 left q 1 frac q 2 q 2 right v v 2 Substituting v u u displaystyle v u u it follows that u displaystyle u satisfies the linear 2nd order ODE u R x u S x u 0 displaystyle u R x u S x u 0 since v u u u u u u 2 u u v 2 displaystyle v u u u u u u 2 u u v 2 so that u u v 2 v S R v S R u u displaystyle u u v 2 v S Rv S Ru u and hence u R u S u 0 displaystyle u Ru Su 0 A solution of this equation will lead to a solution y u q 2 u displaystyle y u q 2 u of the original Riccati equation Application to the Schwarzian equation EditAn important application of the Riccati equation is to the 3rd order Schwarzian differential equation S w w w w w 2 2 f displaystyle S w w w w w 2 2 f 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 S w displaystyle S w has the remarkable property that it is invariant under Mobius transformations i e S a w b c w d S w displaystyle S aw b cw d S w whenever a d b c displaystyle ad bc is non zero The function y w w displaystyle y w w satisfies the Riccati equation y y 2 2 f displaystyle y y 2 2 f By the above y 2 u u displaystyle y 2u u where u displaystyle u is a solution of the linear ODE u 1 2 f u 0 displaystyle u 1 2 fu 0 Since w w 2 u u displaystyle w w 2u u integration gives w C u 2 displaystyle w C u 2 for some constant C displaystyle C On the other hand any other independent solution U displaystyle U of the linear ODE has constant non zero Wronskian U u U u displaystyle U u Uu which can be taken to be C displaystyle C after scaling Thus w U u U u u 2 U u displaystyle w U u Uu u 2 U u so that the Schwarzian equation has solution w U u displaystyle w U u Obtaining solutions by quadrature EditThe 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 y 1 displaystyle y 1 can be found the general solution is obtained as y y 1 u displaystyle y y 1 u Substituting y 1 u displaystyle y 1 u in the Riccati equation yields y 1 u q 0 q 1 y 1 u q 2 y 1 u 2 displaystyle y 1 u q 0 q 1 cdot y 1 u q 2 cdot y 1 u 2 and since y 1 q 0 q 1 y 1 q 2 y 1 2 displaystyle y 1 q 0 q 1 y 1 q 2 y 1 2 it follows that u q 1 u 2 q 2 y 1 u q 2 u 2 displaystyle u q 1 u 2 q 2 y 1 u q 2 u 2 or u q 1 2 q 2 y 1 u q 2 u 2 displaystyle u q 1 2 q 2 y 1 u q 2 u 2 which is a Bernoulli equation The substitution that is needed to solve this Bernoulli equation is z 1 u displaystyle z frac 1 u Substituting y y 1 1 z displaystyle y y 1 frac 1 z directly into the Riccati equation yields the linear equation z q 1 2 q 2 y 1 z q 2 displaystyle z q 1 2 q 2 y 1 z q 2 A set of solutions to the Riccati equation is then given by y y 1 1 z displaystyle y y 1 frac 1 z where z is the general solution to the aforementioned linear equation See also EditLinear quadratic regulator Algebraic Riccati equation Linear quadratic Gaussian controlReferences Edit 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 Ince E L 1956 1926 Ordinary Differential Equations New York Dover Publications pp 23 25Further reading EditHille 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 amp 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 PressExternal links Edit Riccati equation Encyclopedia of Mathematics EMS Press 2001 1994 Riccati Equation at EqWorld The World of Mathematical Equations Riccati Differential Equation at Mathworld MATLAB function for solving continuous time algebraic Riccati equation SciPy has functions for solving the continuous algebraic Riccati equation and the discrete algebraic Riccati equation Retrieved from https en wikipedia org w index php title Riccati equation amp oldid 1117729267, wikipedia, wiki, book, books, library,