ok well i'm pretty much home and dry in this problem
the aim of this problem is to get the general solution for the ode below..
2u'' - xu' + u = 0 = g(x)
i started to solve it by rearranging the equation..
2u'' + u = xu'
apply fourier transform..
2F(u'') + u^ = g^
(-2k^2)u^ + u^ = g^
u^ [1- 2(k^2)] = g^
u^ = {1/ [1- 2(k^2)]}g^
the problem is, i cant find any of the them in the transform table..