Isn't 0 now an irregular singular point? In fact, it is having rank of 1/2. Until someone comes up with a better approach, let me make this suggestion that you probably won't like at all: Since the rank of the singular point is non-integer, let's for starters, work one with the smallest integer rank of k=1. In that case, we could look at the equation:
it deals with irregular singular points of rank one. Can we use that paper to solve this equation first? I realize that's taking a bunch of steps backwards but sometimes you have to bust down a lot of walls before you lay the first course. :)
May I ask that we change the title of this to "Solving a DE with an irregular singular point"? That would make it much more interesting.
Hi saidMath !
Do you known the hypergeometric functions ?
The equation x3y′′(x)+(ax3+bx2+cx+d)y(x)=0 requieres an high level of special functions.
Even with a simpler equation x2y′′(x)+(ax2+bx+c)y(x)=0, the solutions are expressed as a combination of confluent hypergeometric functions (Kummer and Tricomi functions).
Nevertheless, in some particular cases, with particular values of parameters a, b, c, d, the solutions can be reduced to functions of lower level.
So, the question is : are the parameters a, b, c, d resticted to some particular values ?
If not, do not expect to find the solutions in terms of a combination of a finite number of elementary functions and of usual special functions.