The general solution is the homogeneous solution (to the "= 0" equation) which you already have plus any particular specific solution to your actual equation (with the "+5" part). One specific solution to your actual equation is obviously just u = 5/3 (i.e., u is the constant 5/3). So just add 5/3 to your homogeneous solution to get the general solution.
For the comment you posted to Star's thread, a particular solution is again just y = -beta/alpha (i.e., y is the constant -beta/alpha). So just add this to the homogeneous solution (i.e., the solution you get by setting beta = 0).