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Problem 19 Hard Difficulty

(a) Use Euler's method with each of the following step sizes to estimate the value of $ y(0.4), $ where $ y $ is the solution of the initial-value problem $ y' = y, y(0) = 1. $
(i) $ h = 0.4 $ (ii) $ h = 0.2 $ (iii) $ h = 0.1 $
(b) We know that the exact solution of the initial-value problem in part (a) is $ y = e^x. $ Draw, as accurately as you can, the graph of $ y = e^x, 0 \le x \le 0.4, $ together with the Euler approximations using the step sizes in part (a). (Your sketches should resemble Figure 12, 13, and 14.) Use your sketches to decide whether your estimates in part (a) are underestimates or 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(0.4), $ namely $ e^{0.4}. $ What happens to the errors each time the steps size is halved?

Answer

a)
(i) $y(0.4)=1.4$
(ii) $y(0.4)=1.44$
(iii) $y(0.4)=1.4641$
b) We see that the estimates are underestimates since they are all below the graph of $y=e^{x}$
c) Each time the step size is halved, the error estimate also appears to be halved (approximately).

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Video Transcript

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