Need help solving pseudo-hypergeometric ODE

[pbecker314]

Hi All;

In the course of my research, I need to solve the ODE

x*(1+x)*y''(x) + (3 + A*x + B*x^2)*y'(x) + (C + 4*B*x)*y(x) = 0

It's somewhat similar to the hypergeometric equation, but so far I can't find any closed-form solution. Of course, I can obtain power series solutions using the Frobenius method, but I would very much like to obtain a closed-form solution, even if it's only valid for specific values of the constants A, B, and C. I was thinking that a solution in terms of confluent hypergeometric functions might be possible, but I can't find the required transformation. Any help or suggestions would be appreciated.

Thanks,

Pete Becker
George Mason University

 

[g_edgar]

Maple says two independent solutions are...
$$y \left( x \right) ={{\rm e}^{-Bx}}{\it HeunC} \left( -B,-4+A-B,2,-1/2\,B \left( -8+A-B \right) ,-1/2\,{B}^{2}+1/2\,B \left( -8+A \right) -3/2\,A+C+5,1+x \right)$$
and
$$y \left( x \right) ={{\rm e}^{-Bx}}{\it HeunC} \left( -B,4-A+B,2,-1/2\,B \left( -8+A-B \right) ,-1/2\,{B}^{2}+1/2\,B \left( -8+A \right) -3/2\,A+C+5,1+x \right) \left( 1+x \right) ^{4-A+B}$$

Not really much help, since the Heun C function is defined as solution of a certain differential equation...

 

[pbecker314]

Thanks, I do appreciate the effort! I figured there might be Heun solutions, buy they are not very convenient since there is no global theory for those functions. Also, they need to be computed using a two-term recurrence relation, which is a bit of a pain. Hence my hope for a hypergeometric-type solution, at least in some special cases. Nonetheless, I can proceed with my work based on the power series Frobenius solutions (essentially the Heun functions), so it's not a show-stopper for me, just a bit more work! Thanks again,
Cheers,
Pete