Riccati Differential Equation

There are a number of equations known as the Riccati differential equation. The most common is


(Abramowitz and Stegun 1972, p. 445; Zwillinger 1997, p. 126), which has solutions


where j_n(z) and y_n(z) are spherical Bessel functions of the first and second kinds.

Another Riccati differential equation is


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

Yet another Riccati differential equation is


where w^'=dw/dz (Boyce and DiPrima 1986, p. 87). The transformation


leads to the second-order linear homogeneous equation


If a particular solution w_1 to (4) is known, then a more general solution containing a single arbitrary constant can be obtained from


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


(Boyce and DiPrima 1986, p. 87). This result is due to Euler in 1760.

Explore with Wolfram|Alpha


Abramowitz, M. and Stegun, I. A. (Eds.). "Riccati-Bessel Functions." §10.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 445, 1972.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.

Referenced on Wolfram|Alpha

Riccati Differential Equation

Cite this as:

Weisstein, Eric W. "Riccati Differential Equation." From MathWorld--A Wolfram Web Resource.

Subject classifications