Question

Consider the initial-value problem y' = 2x − 3y + 1, y(1) = 7. The analytic solution is y(x) = 1/9 + 2/3 x + (56/9) e^(−3(x − 1)).

(a) Find a formula involving c and h for the local truncation error in the nth step if Euler's method is used.

(b) Find a bound for the local truncation error in each step if h = 0.1 is used to approximate y(1.5). (Proceed as in this example.)

(c) Approximate y(1.5) using h = 0.1 and h = 0.05 with Euler's method. (Round your answers to four decimal places.)

h = 0.1          y(1.5) ≈______

h = 0.05         y(1.5) ≈______

(d) Calculate the errors in part (c) and verify that the global truncation error of Euler's method is O(h). (Round your answers to four decimal places.) Since y(1.5) =______, the error for h = 0.1 is E_0.1 = ______, while the error for h = 0.05 is E_0.05 = ______. With global truncation error O(h) we expect E_0.1/E_0.05 ≈ 2. We actually have E_0.1/E_0.05 = _______ (rounded to two decimal places).

Answer 1

y(1.5)=2.1856912942 for h=0.05

y(1.5)=1.909816 for h=0.1