Riccati Differential Equation

$\begin{displaymath} y'=P(z)+Q(z)y+R(z)y^2, \end{displaymath}$

(1)

where $y'\equiv dy/dz$ . The transformation

$\begin{displaymath} w \equiv - {y'\over yR(z)} \end{displaymath}$

(2)

leads to the second-order linear homogeneous equation

$\begin{displaymath} R(z)y''-[R'(z)+Q(z)R(z)]y'+[R(z)]^2P(z)y = 0. \end{displaymath}$

(3)

Another equation sometimes called the Riccati differential equation is

$\begin{displaymath} z^2w''+[z^2-n(n+1)]w=0, \end{displaymath}$

(4)

which has solutions

$\begin{displaymath} w=Azj_n(z)+Bzy_n(z). \end{displaymath}$

(5)

Yet another form of ``the'' Riccati differential equation is

$\begin{displaymath} {dy\over dz}=az^n+by^2, \end{displaymath}$

(6)

which is solvable by algebraic, exponential, and logarithmic functions only when $n=-{4m/(2m\pm 1)}$ , 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.