solving ordinary differential equations exercise use backward euler scheme for population growth we consider the ode problem td rn / at some timetn ntwe can approximate the derivative tn by backward differencesee fig tn tn tn tn which leads to rn called the backward euler scheme afind an expression for the in terms of and formulate an algorithm for computing bimplement the algorithm in ain function growth_be(n_ dttfor solving rn with time step (dtcimplement the forward euler scheme in function growth_fe(n_ dttas described in bdcompare visually the solution produced by the forward and backward euler schemes with the exact solution when and make two plotsone with : and one with : filenamegrowth_be py exercise use crank-nicolson scheme for population growth it is recommended to do exercise prior to the present one here we look at the same population growth model td rn / the time derivative tcan be approximated by various types of finite differences exercise considers backward difference (fig )while sect explained the forward difference (fig centered difference is more accurate than backward or forward difference tn tn nc tn td this type of differenceapplied at the point tnc tn tis illustrated geomet rically in fig ainsert the finite difference approximation in the ode rn and solve for the unknown nc assuming is already computed and hence known the resulting computational scheme is often referred to as crank-nicolson scheme bimplement the algorithm in ain function growth_cn(n_ dttfor solving rn with time step (dt