Riccati Differential Equation
There are a number of equations known as the Riccati differential equation. The most common is
![z^2w^('')+[z^2-n(n+1)]w=0](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/RiccatiDifferentialEquation/NumberedEquation1.gif) |
(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 second-order linear homogeneous equation
![R(z)y^('')-[R^'(z)+Q(z)R(z)]y^'+[R(z)]^2P(z)y=0.](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/RiccatiDifferentialEquation/NumberedEquation6.gif) |
(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 first-order
linear equation
![v^'=-[Q(z)+2R(z)w_1(z)]v-R(z)](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/RiccatiDifferentialEquation/NumberedEquation8.gif) |
(8)
|
(Boyce and DiPrima 1986, p. 87). This result is due to Euler in 1760.
REFERENCES:
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."
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