and one for the right-hand side functionsf utd vsivsi ii the equation utmeans setting the two vectors equali the components must be pairwise equal since /we get that implies vsii vsi ir the generalized short notation utis very handy since we can derive numerical methods and implement software for this abstract system and in particular application just identify the formulas in the vectorimplement theseand call functionality that solves the differential equation system programming the numerical methodthe general case in python codethe forward euler step unc un tf un tn /being scalar or vector equationcan be coded as [ + [ndt* ( [ ] [ ]both in the scalar and vector case in the vector caseu[nis one-dimensional numpy array of length holding the mathematical quantity un and the python function must return numpy array of length then the expression [ndt* ( [ ] [ ]is an array plus scalar times an array for all this to workthe complete numerical solution must be represented by two-dimensional arraycreated by zeros((n_t+ + )the first index counts the time points and the second the components of the solution vector at one time point that isu[ ,icorresponds to the mathematical quantity uni when we use only one indexas in [ ]this is the same as [ ,:and picks out all the components in the solution at the time point with index then the assignment [ + becomes correct because it is actually an in-place assignment [ + :the nice feature of these facts is that the same piece of python code works for both scalar ode and system of odesthe ode_fe function for the vector ode is placed in the file ode_system_fe py and was written as followsfrom numpy import linspacezerosasarray import matplotlib pyplot as plt def ode_fe(fu_ dtt)n_t int(round(float( )/dt)