After taking another look at the problem I realized the extra term (When simplified) is simply one of the solutions to the homogenous equation multiplied by a constant.

Instead of giving the general solution as Xg = Xp + C1 * E1 + C2 * E2 Dsolve gives the general solution as Xg = Xp + (C1 - (3/5)) * E1 + C2 * E_2

Therefore I now believe the answers given by Dsolve and wolfram alpha are the exact same

Except that for some reason Dsolve does not "absorb" this extra term into the undetermined constant for that particular eigenfunction. Which is kind of funny but I guess its just an artifact of the process used.

I consider this mystery solved but if anyone wants to validate my suspicion or shed some light on why it ends up like this I would still be interested.

Thanks