solving ordinary differential equations ca centered difference approximation to the derivativeas explored in exercise can be written tn tn tn write up the taylor series for tn around tn and the taylor series for tn taround tn subtract the two seriessolve with respect to tn /identify the finite difference approximation and the error terms on the right-hand sideand write up the leading order error term how is this term compared to the ones for the forward and backward differencesdcan you use the leading order error terms in )-cto explain the visual observations in the numerical experiment in exercise efind the leading order error term in the following standard finite difference approximation to the second-order derivativeu tn tn tn tn tt hint express tn tvia taylor series and insert them in the difference formula filenametaylor_differences pdf exercise use backward euler scheme for oscillations consider ( )-( modeling an oscillating engineering system this  ode system can be solved by the backward euler schemewhich is based on discretizing derivatives by collecting information backward in time more specificallyu tis approximated as tu tt general vector ode ut/where and are vectorscan use this approximation as followsun un un tn / which leads to an equation for the new value un un tf un tn un for general this is system of nonlinear algebraic equations howeverthe ode ( )-( is linearso backward euler scheme leads to system of two algebraic equations for two unknownsun tv un tu ( (