Consider the problem of solving
subject to the initial conditions using a discrete approach where is computed at discrete times, where and
(a) By using Taylor expansions around , derive the centred-difference formula
where the value of should be found.
(b) Find the general solution of and show that this is the discrete version of the corresponding general solution to .
(c) The fully discretized version of the differential equation (1) is
By finding a particular solution first, write down the general solution to the difference equation (2). For the solution which satisfies the discretized initial conditions and , find the error in in terms of only.