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

(1)

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

(2)

where
and
are spherical Bessel functions
of the first and second
kinds.
Another Riccati differential equation is

(3)

which is solvable by algebraic, exponential, and logarithmic functions only when ,
for ,
1, 2, ....
Yet another Riccati differential equation is

(4)

where
(Boyce and DiPrima 1986, p. 87). The transformation

(5)

leads to the secondorder linear homogeneous equation

(6)

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

(7)

where
is a solution to the firstorder linear equation

(8)

(Boyce and DiPrima 1986, p. 87). This result is due to Euler in 1760.
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References
Abramowitz, M. and Stegun, I. A. (Eds.). "RiccatiBessel 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: McGrawHill,
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, 343355, 2002.Glaisher,
J. W. L. "On Riccati's Equation." Quart. J. Pure Appl. Math. 11,
267273, 1871.Goldstein, M. E. and Braun, W. H. Advanced
Methods for the Solution of Differential Equations. NASA SP316. Washington,
DC: U.S. Government Printing Office, pp. 4546, 1973.Ince, E. L.
Ordinary
Differential Equations. New York: Dover, pp. 2335 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: McGrawHill,
pp. 6263, 1972.Zwillinger, D. (Ed.). CRC
Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 414,
1995.Zwillinger, D. "Riccati Equation1 and Riccati Equation2."
§II.A.75 and II.A.76 in Handbook
of Differential Equations, 3rd ed. Boston, MA: Academic Press, pp. 121
and 288291, 1997.Referenced on WolframAlpha
Riccati Differential Equation
Cite this as:
Weisstein, Eric W. "Riccati Differential Equation."
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