solving ordinary differential equations we just insert the approximation ( to get unc un un un ( we now assume that un and un are already computed and that unc is the new unknown solving with respect to unc gives unc un un un ( major problem arises when we want to start the scheme we know that but applying ( for to compute leads to ( where we do not know the initial condition can help us to eliminate and this condition must anyway be incorporated in some way to this endwe discretize by centered differenceu it follows that and we can use this relation to eliminate in ( ) ( with and computed from ( )we can compute and so forth from ( exercise asks you to explore how the steps above are modified in case we have nonzero initial condition remark on simpler method for computing we could approximate the initial condition by forward differenceu leading to then we can use ( for the coming time steps howeverthis forward difference has an error proportional to twhile the centered difference we used has an error proportional to which is compatible with the accuracy (error goes like used in the discretization of the differential equation the method for the second-order ode described above goes under the name stormer' method or verlet integration it turns out that this method is mathematically equivalent with the euler-cromer scheme (!or more preciselythe general formula ( is equivalent with the euler-cromer formulabut the scheme for the