
The Riccati differential equation is a very flexible differential equation because it contains a lot of parameters to model a particular problem, contains many functions as special cases, and still retains a structure that is not too complicated and resemblant of a polynomial. Among the solutions of the Riccati Equation are Bessel Equations of the first and second kind, trigonometric functions, elliptic curves, and ratios thereof (see below for several examples of special cases). The Riccati differential equation is:
If alpha is zero, the Riccati differential equation is not a differential equation anymore as the only term containing y' vanishes. The resulting solutions of the remaining equation are polynomials and square roots. If additionally to alpha=0, gamma is different from zero, there are always two solutions. One of them always goes through (0,0), and the other does not go through (0,0). The webMathematica panel below shows both (as continued fraction expansions and in the plots). The continued fraction approximation of the solution of the Riccati Equation that goes through the origin (if it exists) is (using 6 terms):
The continued fraction approximation of the solution of the Riccati Equation that does not go through the origin (if it exists) is (using 6 terms):
The Riccati differential equation contains the following special cases:

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