Numerical Analysis: Is Crank-Nicholson (modified Euler) or Euler more accurate?

Hi All,

Attached is 1) a) and b), with my solution for 1) a), which I would appreciate checking (I know the k should be a 0 for the initial conditions).

Stuck on b) though, with the following being what I know...

"Which one is the most accurate?"

Crank-Nicholson (modified Euler) is more accurate than Euler.

"Can you see this from the expression of LE for the ODE (1)?"

I know smaller mesh size leads to a more accurate result, and that local error in Euler is O(h^{2}), with global error being O(h^{2}) x (K/h) = O(h), where K/h being the width. Also know it is due to truncation.

Not convinced either of the above, even if correct, are sufficient explanation though. And I am struggling to evaluate LE for either. I know the mesh size is 0.25 from (a), and that I can differentiate f(x,y) twice.

Not sure how to get Xn and Xn+1, which I think I need to know to evaluate LE for each.

Euler is Yn+1 = Yn + HF(Xn, Yn).

Modified Euler is Yn+1 = Yn + 1/2(k1 + k2)

I would appreciate some help though, as I am quite stuck. No idea where that theta even comes from...

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