the odespy package offers this method as odespy backward step the purpose of this exercise is to compare three methods and animate the three solutions the backward euler method with : the backward -step method with : the backward -step method with : choose the model problem from sect filenamerod_be_vs_b step py exercise explore adaptive and implicit methods we consider the same problem as in exercise now we want to explore the use of adaptive and implicit methods from odespy to see if they are more efficient than the forward euler method assume that you want the accuracy provided by the forward euler method with its maximum value since there exists an analytical solutionyou can compute an error measure that summarizes the error in space and time over the whole simulations xx uin uni / xt hereuin is the exact solution use the odespy package to run the following implicit and adaptive solvers backwardeuler backward step rkfehlberg experiment to see if you can use larger time steps than what is required by the forward euler method and get solutions with the same order of accuracy hint to avoid oscillations in the solutions when using the rkfehlberg methodthe rtol and atol parameters to rkffehlberg must be set no larger than and respectively you can print out solver_rkf t_all to see all the time steps used by the rkfehlberg solver (if solver is the rkfehlberg objectyou can then compare the number of time steps with what is required by the other methods filenameground_temp_adaptive py exercise investigate the rule athe crank-nicolson method for odes is very popular when combined with diffusion equations for linear ode au it reads unc un aun aunc apply the crank-nicolson method in time to the ode system for onedimensional diffusion equation identify the linear system to be solved