The Method Of Frobenius

Just a pictorial comment. It's always nice to see what the outcome of a differential equation looks like.

And, by the way, it does not seem much of an effort to solve it numerically, for example as in the Pascal snippet below. The solution looks like a sine with decreasing frequency, as has to be expected. The MUCH parameter has been given two different values for checking convergence.
procedure test(much : integer);
var
  y : array of double;
  x,dx : double;
  k : integer;
begin
  SetLength(y,much+1);
  dx := (xmax-xmin)/much;
  y[0] := 0;
  y[1] := y[0] + dx;
  for k := 1 to much-1 do
  begin
    x := k*dx;
    y[k+1] := 2*y[k]-y[k-1]-sqr(dx)/x*y[k];
  end;
end;