The present paper deals with families of non-trivial solutions of the equation ( d dξ w)2 = P w4(ξ)+Q w2(ξ)+R.Onthebasisofthesesolutions,adirectandgeneralizedalgebraic algorithm is described for constructing the new solutions of some nonlinear partial differentialequations(NLPDEs).Subsequently,manynewandmoregeneralexactsolutions intermsoftheWeierstrassellipticfunction ℘(ξ;g2,g3) areobtained.Themethodcanbe appliedtootherNLPDEsinmathematicalphysics.