This is Maple's attempt at solving for c in Equation (14)

> restart;

> s:=diff(Xi(xi),xi);

> t:=diff((s*xi),xi);

> eq1:=(1/xi)*t+(c^2)*(1-xi^2)*Xi(xi)=0;

This is Equation (14), I'm applying only the first boundary condition here.

> p:=dsolve({eq1},Xi(xi));

This is the solution following the imposition of the boundary condition at xi=1.

> assign(p);Xi:=unapply(Xi(xi),xi);

> eq2:=diff(Xi(xi),xi)=0;

> xi:=1;

> eq2;

> simplify(eq2);

> f:=solve(eq2,c);

> _Z:=1;

> f;

Error, (in collect) cannot collect, 1

> h:=solve(eq2,_C2);

So by solving for c we get some mysterious variable _Z that I don't know anything about.
 
 

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