This work deals with the problem of obtaining numerical approximation to solutions of a system of ordinary differential equations and partial differential equations by backward constraints using successive relaxation (Khan, 1992). A feature of the backward constraints idea is that a multistep method should be chosen so that there is a wide separation of the moduli of the roots of the characteristic equations. This, with the application of the appropriate number of artificial conditions initially and terminally ensures stability. The research presented is an attempt to investigate how well backward constraints method performs as an integrator for ordinary and partial differential equations. Butler (Butler, 1986) has shown that unstable multistep schemes can be successively employed for the solution of initial value problem if they are used in conjunction with suitable backward artificial constraints. In Butler (Butler, 1987) it is shown that under conditions that can easily be achieved, the backward constraints method is stable and convergent. We have shown that the backward constraints method is stable and highly accurate for a method of lines solution of model linear partial differential equations.