Dynamic vibration equations are of great importance not only for un-derstanding the dynamic motion of structures, but also for providing knowledge of differential equations to mathematicians. Although sev¬eral studies have been done on dynamic vibration equations, none of the studies considered dynamic vibration equations with .f ( x) = sech x as a restoring force. This study focuses on numerical solutions of conserva¬tive autonomous dynamic vibration equation x = sech x which is time vibration of the di placement of a structure due to the internal forces, with no damping or ex ernal forcing. The objectives of this study were: to determine the numerical solution of the dynamic vibration equation x = sech x u inz _ - ewmark and Theta numerical algorithms; and test the stability of the numerical schemes employed. The Numerical results us¬ing ewmark and Theta methods have been obtained, tabulated using R studio and represented graphically using ggplot. The stability of the algorithm used have also been discussed and the results of our study indicate tha both ::'\ ewrnark and Theta algorithms exhibit stable cases for the olu ion of the softening spring, equation x = - f ( x) when pa¬rameter cha en are very close to the maximum accuracy parameters, otherwi e un stable. This study is of importance to mathematicians and Mechanical Enzineers as the solutions may be applicable in modelling and predicting potential vibration problems and solutions usually not obvious in preliminary engineering designs. This may take advantage of bene¬fits of relative mechanical motion and to resonate systems since design changes prior to manufacture are less expensive and more effective than design modification clone later on.