Question: Problem 3. (10 POINTS) The modified Euler method for the initial value problem y' = f(t, y), y(to) = yo is the following numerical scheme:

Problem 3. (10 POINTS) The modified Euler method for the initial value problem y' = f(t, y), y(to) = yo is the following numerical scheme: Un+1 = Un + Atf tn + ; At, yn + , Atf(tn, Un) ) . (3) Show that if f is linear in both t and y, that is if f (t, y) = at + By + 7, where a, B, y are constants, then the modified Euler method (3) is identical to the Heun method: At Un+1 = yn + 2 If ( tn, Un) + f ( tn+1, Un + Atf (tn, yn))] ( 4)

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