Hi, I would like to solve two differential equations using MATHEMATICA. The first equation is:
(1-x^2)*y''[x] - 2*x*y'[x] + (lambda + gamma^2 *(1-x^2))*y[x]=0
where x is a real variable, 0 <= x <=1, and gamma^2 is formally a complex variable having Re{gamma^2} >= 0, but for the purpose of my planned calculations I can assume that gamma^2 is real and nonnegative. Hence, gamma would be real and nonnegative. Parameter lambda is also assumed to be real. The second equation is:
(1+x^2)*y''[x] + 2*x*y'[x] - (lambda + gamma^2 *(1+x^2))*y[x]=0
where x is a real variable, x>=0, and parameters lambda and gamma^2 are the same as in the first equation. The general solutions of the both equations are likely to be expressed by some spheroidal wave functions, which are implemented in MATHEMATICA, although it is not clear which of these functions would occur here. However, function DSolve[] does not recognise the above equations as related to the spheroidal wave functions and does not return any useful solution. Replacement of lambda by SpheroidalEigenvalue[n,0,gamma] does not help. I would appreciate an advice how to use MATHEMATICA to obtain the MATHEMATICA expressions for the general solutions of the above two equations under assumptions stated. Leslaw
