|
(1) |
where
. The transformation
|
(2) |
leads to the second-order linear homogeneous equation
|
(3) |
Another equation sometimes called the Riccati differential equation is
|
(4) |
which has solutions
|
(5) |
Yet another form of ``the'' Riccati differential equation is
|
(6) |
which is solvable by algebraic, exponential, and logarithmic functions only when
, for ,
1, 2, ....
References
Abramowitz, M. and Stegun, C. 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.
Glaisher, J. W. L. ``On Riccati's Equation.'' Quart. J. Pure Appl. Math. 11, 267-273, 1871.
© 1996-9 Eric W. Weisstein
1999-05-25