remark on the world population the number of people on the planet follows the model / where the net reproduction tvaries with time and has decreased since its top in the current world value of is %and it is difficult to predict future values at the momentthe predictions of the world population point to growth to billion before declining this example shows the limitation of differential equation modelwe need to know all input parametersincluding /in order to predict the future it is seldom the case that we know all input parameters sometimes knowledge of the solution from measurements can help estimate missing input parameters verificationexact linear solution of the discrete equations how can we verify that the programming of an ode model is correctthe best method is to find problem where there are no unknown numerical approximation errorsbecause we can then compare the exact solution of the problem with the result produced by our implementation and expect the difference to be within very small tolerance we shall base unit test on this idea and implement corresponding test function (see sect for automatic verification of our implementation it appears that most numerical methods for odes will exactly reproduce solution that is linear in we may therefore set at and choose any whose derivative is the choice utd is very simplebut we may add anything that is zeroe utd at // this is valid utfor any aband the corresponding ode looks highly non-trivialhoweveru at // using the general ode_fe function in ode_fe pywe may write proper test function as follows (in file test_ode_fe_exact_linear py)def test_ode_fe()"""test that linear ( )= * + is exactly reproduced ""def exact_solution( )return * def (ut)ode return ( exact_solution( ))** -