Firstly, I must admit I cannot see why on earth someone would want to solve anything (...other than a two-point BVP...) by using the Green's function method.
Ok, so we are looking for a symmetric function to satisfy
where I took the liberty of setting [0,2] as the domain, but this is inessential as we can modify the Green's function and get the domain as large as we like.
For , we have , so solves the IVP. and we have the liberty to define G any way we like.
For , the IVP has the solution . So we demand
Since G has to be symmetric, write
and observe that (1) is satisfied for . Define G this way on . So we get the solution of the IVP as