solving ordinary differential equations numerical solution we have not looked at numerical methods for handling second-order derivativesand such methods are an optionbut we know how to solve first-order differential equations and even systems of first-order equations with littleyet very commontrick we can rewrite ( as first-order system of two differential equations we introduce and as two new unknown functions the two corresponding equations arise from the definition and the original equation ( ) ( ( (notice that we can use to remove the second-order derivative from newton' nd law we can now apply the forward euler method to ( )-( )exactly as we did in sect unc un vn nc un ( ( resulting in the computational scheme unc un nc programming the numerical methodthe special case simple program for ( )-( follows the same ideas as in sect from numpy import zeroslinspacepicosarray import matplotlib pyplot as plt omega *pi/omega dt / * n_t int(round( /dt) linspace( n_t*dtn_t+ zeros(n_t+ zeros(n_t+ initial condition x_ [ x_ [ ( (