- Riccati equation
In
mathematics , a Riccati equation is anyordinary differential equation that has the form:It is named after CountJacopo Francesco Riccati (1676-1754).Reduction to a second order linear equation
As explained on pages 23-25 of Ince's book, the non-linear Riccati equation can always be reduced to a second order linear
ordinary differential equation (ODE). Indeed if:then, wherever is non-zero, satisfies a Riccati equation of the form:where and .In fact: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
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 solutionObtaining solutions by quadrature
The correspondence between Riccati equations and 2nd 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 can find one particular solution , the general solution is obtained as:Substituting:in the Riccati equation yields:and since::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.
External links
* [http://eqworld.ipmnet.ru/en/solutions/ode/ode0123.pdf Riccati Equation] at EqWorld: The World of Mathematical Equations.
* [http://mathworld.wolfram.com/RiccatiDifferentialEquation.html Riccati Differential Equation] atMathworld Bibliography
*cite book|last=Hille|first=Einar|title=Ordinary Differential Equations in the Complex Domain|year=1997|origyear=1976|publisher=Dover Publications|location=New York|id=ISBN 0-486-69620-0
*cite book|last=Ince|first=E. L.|title=Ordinary Differential Equations|year=1956|origyear=1926|publisher=Dover Publications|location=New York
*cite book|last=Nehari|first=Zeev|title=Conformal Mapping|year=1975|origyear=1952|publisher=Dover Publications|location=New York|id=ISBN 0-486-61137-X
*cite book|last=Polyanin|first=Andrei D.|coauthors=and Valentin F. Zaitsev|title=Handbook of Exact Solutions for Ordinary Differential Equations|year=2003|edition=2nd ed.|publisher=Chapman & Hall/CRC|location=Boca Raton, Fla.|id=ISBN 1-58488-297-2
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