Question

(a) Use Euler's method with each of the following step sizes to estimate the value of y(0.8), where y is the solution of the initial-value problem

y' = y, y(0) = 3.

(i)    h = 0.8

y(0.8) =

(ii)    h = 0.4

y(0.8) =

(iii)    h = 0.2

y(0.8) =

(b) We know that the exact solution of the initial-value problem in part (a) is

y = 3e^x.

Draw, as accurately as you can, the graph of

y = 3e^x,

0 ≤ x ≤ 0.8

together with the Euler approximations using the step sizes in part (a). (Your sketches should resemble these figures.) Use your sketches to decide whether your estimates in part (a) are underestimates or overestimates.

The estimates are  ---Select--- underestimates overestimates .

(c) The error in Euler's method is the difference between the exact value and the approximate value. Find the errors made in part (a) in using Euler's method to estimate the true value of y(0.8), namely

3e^0.8.

(Round your answers to four decimal places.)

h = 0.8	error = (exact value) − (approximate value) =
h = 0.4	error = (exact value) − (approximate value) =
h = 0.2	error = (exact value) − (approximate value) =

What happens to the error each time the step size is halved?

Each time the step size is halved, the error estimate appears to be  ---Select--- halved reduced to a third tripled doubled unchanged (approximately).

Answer 1

Ruler Method orn r00 3- 8 0=4 ニ3 3 0.8Ca) =3 +2-4 ニ54 -3+o4 (3) 3+(2 ニ42

,=4o+h fa, o) 3+I'2 4 '2 9a-9,th fa,, 42 +(0.4) C4.2) 5.88 Cin) thesa) heo2 fMoro) -3 = 3t012)C3) - 3.6 316-t(012) C3.6)

an22 14.32 + 4.32+Co.2) C4 9 584 Tasia)518 + 5.184 t0)( 5.184) 6.22 08 A otual uaue yo>3 Trans

5.184 100a)(5.184) &.22 08 A ctual uaCue Yo>3 y = Be y 6.64662 5u4-12 7662 Eror 6.6766 8.0my o.7662 ENror - 6.64662-5.8 he.y Enror 6646 2 - 6.2208o: 45582 nC.