Riccati equation

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In mathematics, a Riccati equation is any ordinary differential equation that has the form

$y' = q_0(x) + q_1(x) \, y + q_2(x) \, y^2$

It is named after Count Jacopo Francesco Riccati (1676-1754).

1 Reduction to a second order linear equation
2 Application to the Schwarzian equation
3 Obtaining solutions by quadrature
4 Real World Applications
5 External link
6 Bibliography

[edit] 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

y' = q 0 (x) + q 1 (x) y + q 2 (x) y 2

then, wherever $q 2$ is non-zero, $v = y q 2$ satisfies a Riccati equation of the form

v' = v 2 + P (x) v + Q (x),

where $Q = q 2 q 0$ and $P = q 1 + (q 2' / q 2)$ . In fact

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 .

Substituting $v = - u' / u$ , it follows that $u$ satisfies the linear 2nd order ODE

u'' - P (x) u' + Q (x) u = 0

since

v' = - (u' / u)' = - (u'' / u) + (u' / u) 2 = - (u'' / u) + v 2

so that

u'' / u = v 2 - v' = - Q - P v = - Q + P u' / u

and hence

u'' - P u' + Q u = 0.

A solution of this equation will lead to a solution $y = - u' / (q 2 u)$ of the original Riccati equation.

[edit] 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. $S (a w + b / c w + d) = S (w)$ whenever $a d - b c$ is non-zero.) The function $y = w'' / w'$ satisfies the Riccati equation

y' = y 2 / 2 + f .

By the above $y = - 2 u' / u$ where $u$ is a solution of the linear ODE

u'' + (1 / 2) f u = 0.

Since $w'' / w' = - 2 u' / u$ , integration gives $w' = C / u 2$ for some constant $C$ . On the other hand any other independent solution $U$ of the linear ODE has constant non-zero Wronskian $U' u - U u'$ which can be taken to be $C$ after scaling. Thus

w' = (U' u - U u') / u 2 = (U / u)'

so that the Schwarzian equation has solution $w = U / u .$

[edit] 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 the 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 $y 1$ , the general solution is obtained as

y = y 1 + u

Substituting

y 1 + u

in the Riccati equation yields

$y_1' + u' = q_0 + q_1 \cdot (y_1 + u) + q_2 \cdot (y_1 + u)^2,$

and since

$y_1' = q_0 + q_1 \, y_1 + q_2 \, y_1^2$

$u' = q_1 \, u + 2 \, q_2 \, y_1 \, u + q_2 \, u^2$

$u' - (q_1 + 2 \, q_2 \, y_1) \, u = q_2 \, u^2,$

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

$z =\frac{1}{u}$

Substituting

$y = y_1 + \frac{1}{z}$

directly into the Riccati equation yields the linear equation

$z' + (q_1 + 2 \, q_2 \, y_1) \, z = -q_2$

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

$y = y_1 + \frac{1}{z}$

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

[edit] Real World Applications

Bass diffusion model.

[edit] External link

Riccati Equation at EqWorld: The World of Mathematical Equations.

[edit] Bibliography

Hille, Einar [1976] (1997). Ordinary Differential Equations in the Complex Domain. New York: Dover Publications. ISBN 0-486-69620-0.
Ince, E. L. [1926] (1956). Ordinary Differential Equations. New York: Dover Publications.
Nehari, Zeev [1952] (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.