The generalizations, due to Kantorovich et al. of the well-known numerical algorithms, successive approximations, steepest decent, and Newton's method; onto normed spaces, Hilbert spaces, and Banach spaces, respectively, have been tested on a variety of equations occuring in engineering and physics. Analytical (nonnumerical) solutions to a second-order partial differential equation, a nonlinear first-order ordinary differential equation, van der Pol's equation, a nonlinear damping problem, and a nonlinear two-point boundary value problem were obtained by symbol manipulation as, for example, provided by FORMAC. These algorithms result in relatively simple forms, e.g., polynomials in sines and cosines, depending on the choice of the initial approximation, and yield high accuracy in a few iterations and in seconds-to-minutes of machine time. It is suggested, on the basis of these experiments, that functional analysis algorithms, as developed by Kantorovich, evaluated by automatic formula manipulation can yield analytical solutions of any desired accuracy to a variety of functional equations. In this way, analytical solutions are obtained providing qualitative information while subsequent numerical evaluation avoids much of the art and inaccuracy associated with numerical procedures.