exercises exercise geometric construction of the forward euler method section describes geometric interpretation of the forward euler method this exercise will demonstrate the geometric construction of the solution in detail consider the differential equation with we use time steps astart at and draw straight line with slope go one time step forward to and mark the solution point on the line bdraw straight line through the solution point tu with slope td go one time step forward to and mark the solution point on the line cdraw straight line through the solution point tu with slope td go one time step forward to and mark the solution point on the line dset up the forward euler scheme for the problem calculate and check that the numbers are the same as obtained in )-cfilenameforwardeuler_geometric_solution py exercise make test functions for the forward euler method the purpose of this exercise is to make file test_ode_fe py that makes use of the ode_fe function in the file ode_fe py and automatically verifies the implementation of ode_fe athe solution computed by hand in exercise can be used as reference solution make function test_ode_fe_ (that calls ode_fe to compute three time steps in the problem uu and compare the three values and with the values obtained in exercise bthe test in acan be made more general using the fact that if is linear in and does not depend on ti we have rufor some constant rthe forward euler method has closed form solution as outlined in sect un rt/ use this result to construct test function test_ode_fe_ (that runs number of steps in ode_fe and compares the computed solution with the listed formula for un filenametest_ode_fe py exercise implement and evaluate heun' method aa nd-order runge-kutta methodalso known has heun' methodis derived in sect make function ode_heun(fu_ dtt(as counterpart to ode_fe(fu_ dttin ode_fe pyfor solving scalar ode problem ut/ with this method using time step size bsolve the simple ode problem uu by the ode_heun and the ode_fe function make plot that compares heun' method and the forward euler method with the exact solution td for oe use time step :