first time level ( implements the initial condition slightly more accurately than what is naturally done in the euler-cromer scheme the latter will do tv which differs from in ( by an amount because of the equivalence of ( with the euler-cromer schemethe numerical results will have the same nice properties such as constant amplitude there will be phase error as in the euler-cromer schemebut this error is effectively reduced by reducing tas already demonstrated the implementation of ( and ( is straightforward in function (file osc_ nd_order py)from numpy import zeroslinspace def osc_ nd_order(u_ omegadtt)""solve 'omega** * for in ( , ] ( )=u_ and '( )= by central finite difference method with time step dt ""dt float(dtnt int(round( /dt) zeros(nt+ linspace( nt*dtnt+ [ u_ [ [ *dt** *omega** * [ for in range( nt) [ + * [nu[ - dt** *omega** * [nreturn ut finite difference methodlinear damping key issue is how to generalize the scheme from sect to differential equation with more terms we start with the case of linear damping term bu possibly nonlinear spring force /and an excitation force /mu bu ud / ( we need to find the appropriate difference approximation to in the bu term good choice is the centered difference tn unc un sampling the equation at time point tn mu tn bu tn un tn /(