The Boole 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 Boole 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 Boole differential equation is:
If alpha is zero, the Boole differential equation is not a differential equation anymore as the only term containing y' vanishes. The resulting solutions of the remaining equation are modular forms and elliptic curves, which are also very important functions in mathematics and very interesting objects to study (they are the key ingredient in the proof to the legendary Fermat's Last Theorem). Although the case alpha=0 doesn't leave us with a differential equation anymore, this case has not been excluded here to allow for the study of the resulting elliptic curves and their continued fraction expansions.
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 Boole Equation that goes through the origin (if it exists) is (using 6 terms):
The continued fraction approximation of the solution of the Boole Equation that does not go through the origin (if it exists) is (using 6 terms):
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