Chapter one includes introduction to dynamic relaxation method (DR) which is combined with finite differences method (FD) for the sake of solving ordinary and partial differential equations, as a single equation or as a group of differential equations. In this chapter the dynamic relaxation equations are transformed to artificial dynamic space by adding damping and inertia effects. These are then expressed in finite difference form and the solution is obtained through iterations. In chapter two the procedural steps in solving differential equations using DR method were applied to the system of differential equations (i.e. ordinary and/ or partial differential equations). The DR program performs the following operations: Reads data file; computes fictitious densities; computes velocities and displacements; checks stability of numerical computations; checks convergence of solution; and checks wrong convergence. At the end of this chapter the dynamic relaxation (DR) numerical method coupled with the finite differences discretization technique is used to solve nonlinear ordinary and partial differential equations. Subsequently, a FORTRAN program is developed to generate the numerical r