The asymptotic solution for the incompressible viscous flow past a semi-infinite flat plate constructed by Goldstein (1956, 1960) can be valid only if the solutions of certain ordinary differential equations obey certain constraints (given in Goldstein 1956, 1960). In this paper, we construct the solutions of these equations, show that the necessary constraints are met, and hence establish the validity of the asymptotic solution up to the order considered. The manner in which undetermined constants appear in the solution are discussed.