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Riccati Differential Equation

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The Riccati differential equation is the first-order nonlinear ordinary differential equation

w^'=P(z)+Q(z)w+R(z)w^2,
(1)

where w^'=dw/dz and P, Q, and R are given functions (Boyce and DiPrima 1986, p. 87). A special form is

(dy)/(dz)=az^n+by^2,
(2)

which is solvable by algebraic functions, exponential functions, and logarithmic functions only when n=-4m/(2m+/-1), for m=0, 1, 2, ....

The transformation

w=-(y^')/(yR(z))
(3)

leads to the second-order linear homogeneous equation

R(z)y^('')-[R^'(z)+Q(z)R(z)]y^'+[R(z)]^2P(z)y=0.
(4)

If a particular solution w_1 to (3) is known, then a more general solution containing a single arbitrary constant can be obtained from

w=w_1(z)+1/(v(z)),
(5)

where v(z) is a solution to the first-order linear differential equation

v^'=-[Q(z)+2R(z)w_1(z)]v-R(z)
(6)

(Boyce and DiPrima 1986, p. 87).

The similarly named Riccati-Bessel functions satisfy a distinct second-order linear differential equation.

See also

Bernoulli Differential Equation, Riccati-Bessel Functions

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References

Bender, C. M. and Orszag, S. A. §1.6 in Advanced Mathematical Methods for Scientists and Engineers. New York: McGraw-Hill, 1978.Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley, 1986.Boyle, P. P.; Tian, W.; and Guan, F. "The Riccati Equation in Mathematical Finance." J. Symb. Comput. 33, 343-355, 2002.Glaisher, J. W. L. "On Riccati's Equation." Quart. J. Pure Appl. Math. 11, 267-273, 1871.Goldstein, M. E. and Braun, W. H. Advanced Methods for the Solution of Differential Equations. NASA SP-316. Washington, DC: U.S. Government Printing Office, pp. 45-46, 1973.Ince, E. L. Ordinary Differential Equations. New York: Dover, pp. 23-35 and 295, 1956.Reid, W. T. Riccati Differential Equations. New York: Academic Press, 1972.Simmons, G. F. Differential Equations with Applications and Historical Notes. New York: McGraw-Hill, pp. 62-63, 1972.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 414, 1995.Zwillinger, D. "Riccati Equation--1 and Riccati Equation--2." §II.A.75 and II.A.76 in Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 121 and 288-291, 1997.

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Riccati Differential Equation

Cite this as:

Weisstein, Eric W. "Riccati Differential Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RiccatiDifferentialEquation.html

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