spheroidal wave equation

In mathematics, the spheroidal wave equation is given by

:$(1-t^2)\backslash frac\{d^2y\}\{dt^2\}\; -2(b+1)\; t\backslash ,\; \backslash frac\{d\; y\}\{dt\}\; +\; (c\; -\; 4qt^2)\; \backslash ,\; y=0$

It is a generalization of the Mathieu differential equation.see Abramowitz and Stegun, [http://people.math.sfu.ca/~cbm/aands/page_722.htm page 722]

If $y(t)$ is a solution to this equation and we define $S(t):=(1-t^2)^\{b/2\}y(t)$, then $S(t)$ is a prolate spheroidal wave function in the sense that it satisfies the equationsee Bateman, [https://archive.org/details/partialdifferent033549mbp/page/n469, page 442]

:$(1-t^2)\backslash frac\{d^2S\}\{dt^2\}\; -2\; t\backslash ,\; \backslash frac\{d\; S\}\{dt\}\; +\; (c\; -\; 4q\; +\; b\; +\; b^2\; +\; 4q(1-t^2)\; -\; \backslash frac\{b^2\}\{1-t^2\}\; )\; \backslash ,\; S=0$