In mathematics, a Riccati equation is any ordinary differential equation that has the form For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...
In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...
It is named after Count Jacopo Francesco Riccati (1676-1754). Jacopo Francesco Riccati (28 May 1676 - 15 April 1754) was an Italian mathematician, from Venice. ...
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 In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...
- y' = q0(x) + q1(x)y + q2(x)y2
then, wherever q2 is non-zero, v = yq2 satisfies a Riccati equation of the form
- v' = v2 + P(x)v + Q(x),
where Q = q2q0 and P = q1 + (q2' / q2). In fact
- v' = (yq2)' = y'q2 + yq2' = (q0 + q1y + q2y2)q2 + vq2' / q2 = q0q2 + (q1 + q2' / q2)v + v2.
Substituting v = − u' / u, it follows that u satisfies the linear 2nd order ODE
- u'' − P(x)u' + Q(x)u = 0
- v' = − (u' / u)' = − (u'' / u) + (u' / u)2 = − (u'' / u) + v2
- u'' / u = v2 − v' = − Q − Pv = − Q + Pu' / u
- u'' − Pu' + Qu = 0.
A solution of this equation will lead to a solution y = − u' / (q2u) 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
- 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) has the remarkable property that it is invariant under Möbius transformations, i.e. S(aw + b / cw + d) = S(w) whenever ad − bc is non-zero.) The function y = w'' / w' satisfies the Riccati equation In mathematics, the Schwarzian derivative is a certain operator that is invariant under all linear fractional transformations. ...
- y' = y2 / 2 + f.
By the above y = − 2u' / u where u is a solution of the linear ODE
- u'' + (1 / 2)fu = 0.
Since w'' / w' = − 2u' / u, integration gives w' = C / u2 for some constant C. On the other hand any other independent solution U of the linear ODE has constant non-zero Wronskian U'u − Uu' which can be taken to be C after scaling. Thus
- w' = (U'u − Uu') / u2 = (U / u)'
so that the Schwarzian equation has solution w = U / u.
Obtaining 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 y1, the general solution is obtained as
- y = y1 + u
- y1 + u
in the Riccati equation yields
which is a Bernoulli equation. The substitution that is needed to solve this Bernoulli equation is In mathematics, an ordinary differential equation of the form is called a Bernoulli differential equation or Bernoulli equation. ...
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 Equation at EqWorld: The World of Mathematical Equations.
- Riccati Differential Equation at Mathworld
MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...
- Hille, Einar  (1997). Ordinary Differential Equations in the Complex Domain. New York: Dover Publications. ISBN 0-486-69620-0.
- Ince, E. L.  (1956). Ordinary Differential Equations. New York: Dover Publications.
- Nehari, Zeev  (1975). Conformal Mapping. New York: Dover Publications. ISBN 0-486-61137-X.
- Polyanin, Andrei D.; and Valentin F. Zaitsev (2003). Handbook of Exact Solutions for Ordinary Differential Equations, 2nd ed., Boca Raton, Fla.: Chapman & Hall/CRC. ISBN 1-58488-297-2.