understanding the forward euler method the good thing about the forward euler method is that it gives an understanding of what differential equation is and geometrical picture of how to construct the solution the first idea is that we have already computed the solution up to some time point tn the second idea is that we want to progress the solution from tn to tnc as straight line we know that the line must go through the solution at tn the point tn un the differential equation tells us the slope of the lineu tn un tn run that isthe differential equation gives direct formula for the further direction of the solution curve we can say that the differential equation expresses how the system (uundergoes changes at point there is general formula for straight line ax with slope that goes through the point / using this formula adapted to the present caseand evaluating the formula for tnc results in unc run tnc tn un un run which is nothing but the forward euler formula you are now encouraged to do exercise to become more familiar with the geometric interpretation of the forward euler method programming the forward euler schemethe general case our previous program was just flat main program tailored to special differential equation when programming mathematicsit is always good to consider (largeclass of problems and making python function to solve any problem that fits into the class more specificallywe will make software for the class of differential equation problems of the form td ut/ oe for some given function and numbers and we also take the opportunity to illustrate what is commonly called demo function as the name impliesthe purpose of such function is solely to demonstrate how the function works (not to be confused with test functionwhich does verification by use of assertthe python function calculating the solution must take tand as inputfind the corresponding compute the solutionand return an array with un and an array with tn the forward euler scheme reads unc un tf un tn / the corresponding program may now take the form (file ode_fe py)