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

It is named after Count Jacopo 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

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

since

v' = - (u' / u)' = - (u'' / u) + (u' / u)² = - (u'' / u) + v²

so that

u'' / u = v² - v' = - Q - Pv = - Q + Pu' / u

and hence

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

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

Substituting

y₁ + u

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 link

• Riccati Equation at EqWorld: The World of Mathematical Equations.
• Riccati Differential Equation at Mathworld

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. 

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