Chapter 8.3, Problem 18P

Consider the improved Euler method for solving theillustrative initial value problem $\begin{array}{cc}y\text{'}=1-t+4y,& y(0)=1\end{array}$. Using the result of Problem 17(c) and the exact solution of theinitial value problem, determine $e_{n+1}$ and a bound for the error at any step on $0\le t\le 2$.

Compare this error with the oneobtained in Eq. (25) of Section 8.2 using the Euler method.

Also obtain a bound for $e_1$ for $h=0.05$, and compare it with Eq.(26) of Section 8.2.