Chapter 9.2, Problem 19E

(a) Use Euler’s method with each of the following step sizes to estimate the value of $y\left(0.4\right)$, where y is the solution of the initial-value problem $y^{\prime }=y,y\left(0\right)=1$.

(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 Figures 11, 12, and 13.) 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 $y\left(0.4\right)$, namely, $e^{0.4}$. What happens to the error each time the step size is halved?