(a) We start with x=0, y=1, y'=-y=-1. So, dy = -dx = -0.1.
Therefore, we approximate y(0.1) as 1-0.1 = 0.9.
(b) I don't know any book formulas for error bounds in Euler's method. What formula have you been given? Have Taylor Series been suggested?
Perhaps I could say that y''=-y'=y. Since y'<0, we have (at least for small x) that |y''| = |y'| = |y|, a decreasing quantity somewhere between 0.9 and 1, at least for 0<x<0.1.
Since 0.9<y''<1, we have 0.9x-1 < y' < x-1 (by integrating and using initial condition).
Continuing in the same way, we have:
0.45x^2 - x + 1 < y < 0.5x^2 - x + 1.
Evaluating these endpoints at 0.1, we find
0.9045 < y < 0.905. As we want to know the distance from y to 0.9, we would say 0.0045 < y-0.9 < 0.005. Usually we aren't so concerned with error lowerbounds, so I would just take 0.005 as an error upperbound.
(c) The actual error is e^(-0.1)-0.9, or about 0.0048, which is nicely nestled in the error window I identified earlier...but most importantly, it is constrained by the upperbound I gave.
Edit: schmiso, the O(h^3) error term in your reference leaves me wondering how reliable your one-term error estimate is as an error bound. Perhaps part (b) should be asking for an estimate, rather than a bound?