Page - 142 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

142 4 SolvingOrdinaryDifferentialEquations Fig.4.25 General oscillatingsystem Fig.4.26 Apendulumwith forces Thisequation is, however,morecommonlyreordered to mu00Cf.u0/Cs.u/DF.t/: (4.66) Because the differential equation is of second order, due to the termu00, we need two initial conditions: u.0/DU0; u0.0/DV0: (4.67) Note thatwith the choicesf.u0/ D 0, s.u/ D ku, andF.t/ D 0we recover the originalODEu00C!2uD0with!Dpk=m. Howcanwe solve (4.66)? As for the simpleODEu00 C!2u D 0, we start by rewriting the second-orderODEasasystemof twofirst-orderODEs: v0 D 1 m .F.t/ s.u/ f.v//; (4.68) u0 Dv: (4.69) The initial conditionsbecomeu.0/DU0 andv.0/DV0.