The present paper deals with families of non-trivial solutions of the equation (ddξw)2=Pw4(ξ)+Qw2(ξ)+R. On the basis of these solutions, a direct and generalized algebraic algorithm is described for constructing the new solutions of some nonlinear partial differential equations (NLPDEs). Subsequently, many new and more general exact solutions in terms of the Weierstrass elliptic function ℘(ξ;g2,g3)℘(ξ;g2,g3) are obtained. The method can be applied to other NLPDEs in mathematical physics.