Hey nyson13.

Just to clarify, is c2 = c1 + 1? I'll assume this is the case right now (from the context of your post).

The basic Euler equation is u(x+a) = u(a) + u'(a)(x-a), but you actually have two derivative terms so I'll go off the basis that you are going to use two Euler expansions and then collect the terms. Let a be c1 and b be c2 where u(x+b) = u(b) + u'(b)(x-b). (These are of course approximations not real equalities). Take (2) - (1) and we get:

u(x+b) - u(x+a) = u(b) - u(a) + u'(b)(x-b) - u'(a)(x-a).

= u(b) - u(a) + x*(u'(b) - u'(a)) + u'(a)*a - u'(b)*b

if b = a + 1 then this means

= u(a+1) - u(a) + x*(u'(a+1) - u'(a)) + u'(a)*a - u'(a+1)*(a+1)

If x = 0 we get cancellations:

u(a+1) - u(a) = u(a+1) - u(a) + 0 + u'(a)*a - u'(a+1)*(a+1) which gives

u'(a)*a - u'(a+1)*(a+1) = 0 which gives

u'(a)*a = (1+a)u'(a+1) or

u'(a) = [1+a]/a * u'(a+1) or

u'(a) = [1/a + 1]*u'(a+1) with the Euler approximation.

Now do you have values for u'(a) and for u'(a+1) in terms of u's and a's (your notation is a little confusing) and does it match this? (Or have I screwed up somewhere along the lines?)