Frequently, diffusion processes require the determination of a free surface from overprescribed boundary data. A commonly used constructive solution technique for such problems is the method of straight lines. This paper illustrates the steps involved in the solution process. Specifically, the method of lines is used to approximate various explicit and implicit free boundary problems for a linear one-dimensional diffusion equation with a sequence of free boundary problems for ordinary differential equations. It is shown that these equations have solutions which can be readily obtained with the method of invariant imbedding. It also is established for a model problem that the approximate solutions converge to a unique (almost) classical solution as the discretization parameter goes to zero.