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Riccati Differential Equation

\end{displaymath} (1)

where $y'\equiv dy/dz$. The transformation
w \equiv - {y'\over yR(z)}
\end{displaymath} (2)

leads to the second-order linear homogeneous equation
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
\end{displaymath} (4)

which has solutions
\end{displaymath} (5)

Yet another form of ``the'' Riccati differential equation is
{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 $m=0$, 1, 2, ....


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