   Something to chew on for the math savy
  I've been pondering on a nonlinear second degree diff equation. It seems simple enough but after a few hours of trying I still havent found any solutions. Anyways, I've got to take a break now to go bouldering but I thought I might throw it out here if anyone wants to have a go at it:f"(x) + x-1f'(x) - k2f(x) = 0(since its not game related I decided to post it here instead of the math forum)   
  I believe you can obtain the solution by just taking the laplace transform of the 2nd order homogenous equation, and then working the algebra and doing the inverse transform. I should know how to do this, so I'll brush up on my Laplace Transforms later today. Here is what I have so far:y'' + y'/x - k^2*y = 0Now take the laplace  transform:[s^2*X(s) - s*X(0) - X'(0)] + [integral(X'(z) dz, s, infinity)] - k^2*X(s) = 0Now solve for X(s). Then do the inverse laplace transform!Note that you will have constants emerging for the y(0) and the y'(0).   ;  Here is what I got using Laplace transform:f"(x) + x-1f'(x) - k2f(x) = 0=> xf"(x) + f'(x) - xk2f(x) = 0let L{f(x)} = F(z)L{-k2xF} = k2F'(z)L{f'(x)} = zF(z) - f(0)L{xf"(x)} = -(L{f"(x)})' = -(z(zF(z) - f(0)) - f'(0))' = -2zF(z) + z2F'(z) - f(0)so L{xf"(x) + f'(x) - k2xf(x)} = L{0}-2zF(z) + z2F'(z) - f(0) + zF(z) - f(0) + k2F'(z) = 0(z2 + k2)F'(z) - zF(z) - 2f(0) = 0A nonlinear first order diff equation. Might be easier to solve.[edit]Yep, the laplace transformed one has a really easy solution:F(z) = 2xk-2f(0)Hopefully there are no mistakes so the transform back will be a solution.[edit 2]The inverse transform is the derivative of the delta distribution. Not much of an answer really, maybe there are no solutions.[Edited by - Jesper T on February 1, 2008 1:08:49 PM];  I plugged it into maple 11>    f(x) = _C1 BesselI(0, k x) + _C2 BesselK(0, k x)where BesselI is:BesselI and BesselK are the modified Bessel functions of the first and second kinds, respectively. They satisfy the modified Bessel equation: 2 2    2    x  y'' + x y' - (x  + v ) y = 0Does this help at all? I have no experience with the Bessel functions...   ;  Yeah, that makes sense. The bessel functions are a basis for the function space so this is should be a general solution. Seems like maple is a pretty handy program.   