solving ordinary differential equations from numpy import linspacezerosexp import matplotlib pyplot as plt def ode_fe(fu_ dtt)n_t int(round(float( )/dt) zeros(n_t+ linspace( n_t*dtlen( ) [ u_ for in range(n_t) [ + [ndt* ( [ ] [ ]return ut def demo_population_growth()"""test caseu'= *uu( )= ""def (ut)return * ut ode_fe( =fu_ = dt= = plt plot(tut *exp( * )plt show(if __name__ ='__main__'demo_population_growth(this program filecalled ode_fe pyis reusable piece of code with general ode_fe function that can solve any differential equation utand demo function for the special case : uu observe that the call to the demo function is placed in test block this implies that the call is not active if ode_fe is imported as module in another programbut active if ode_fe py is run as program the solution should be identical to what the growth py program produces with the same parameter settings ( : this feature can easily be tested by inserting print statementbut much betterautomated verification is suggested in exercise you are strongly encouraged to take "breakand do that exercise now remark on the use of as variable in the ode_fe programthe variable is used in different contexts inside the ode_fe functionu is an arraybut in the ( ,tfunctionas exemplified in the demo_population_growth functionthe argument is number typicallywe call (in ode_fewith the argument as one element of the array in the ode_fe functionu[nmaking the population growth model more realistic exponential growth of population according the model rn with exponential solution rt is unrealistic in the long run because the resources needed to feed the population are finite at some point there will not be enough resources and the growth will decline common model taking this effect into account as