A solution of the limitation of Calculus in terms of not being able to establish some form of a unified theory of analytical integration was achieved by introducing an entirely new computational algorithm called specialized differential forms or (SDF) that involve the use of Multivariate Polynomials and the differential of Multivariate Polynomials, all defined in a very unique algebraic configuration. Such a unique algorithm would lead to some form of a unified theory of analytical integration that would be driven entirely by computation. This new universal theory in mathematics would be applicable for solving those DEs whose exact solution can only be expressed in terms of the algebraic and elementary functions. Concrete "numerical evidence" of this will be shown on a number of very simple mathematical models involving a first order ODE and a second order PDE. A series of Physical models  have been carefully chosen just for demonstrating the applicability of the current unified theory of integration into the Physical Sciences. These will include the PDEs used for describing a very specific case of the Navier-Stokes equations corresponding to an incompressible fluid involving heat transfer and variable viscosity. To fundamentally illustrate the complete versatility of our unique algorithm for solving PDEs, we would be describing a general process for attempting to solve the set of PDEs that govern general linear elastic boundary value problems.