Yes, you are right. Both II and III are true. I is just meaningless- there is no reason to think that "x3" is actually the number "3".
Suppose Euler’s method, with increment dx, is used to
numerically solve the differential equation dy/dx=f(x,y)
with the initial condition that (x0, y0) lies on the solution
curve. Let (x1, y1), (x2, y2), and so on denote the points
generated by Euler’s method, and let y = y(x) denote the
exact solution to the initial value problem. Which of the
following must be true?
Note: more than one are possible
I. y(3) = y(x3)
II. y2 = y1 + f (x1, y1) dx
III. x3 = x0 + 3 dx
My work:
I believe it is II and III.
III seems correct because dx is the change in x so if you multiply it by 3 and add it to the original it should give you x3. and II just looks like the Euler's formula.
Anyone willing to confirm or correct.