Chapter 8.1, Problem 23P

(a)

To determine

To show: That $\left|E_{n+1}\right|\le \left|E_n\right|+h\left|f\left(t_n,\varphi \left(t_n\right)\right)-f\left(t_n,y_n\right)\right|+\frac{1}{2}{h}^2\left|{\varphi }^{″}\left(t_n\right)\right|\le \alpha \left|E_n\right|+\beta h^2$ where $\alpha =1+hL\text{ and }\beta =\max \frac{\left|{\varphi }^{″}\left(t\right)\right|}{2}\text{ on }{t}_0\le t\le t_n$

(b)

To determine

To show: That $\left|E_n\right|\le \frac{{\left(1+hL\right)}^n-1}{L}\beta h$, given that if $E_0=0\text{ and }\left|E_n\right|$ satisfies $\left|E_{n+1}\right|\le \left|E_n\right|+h\left|f\left(t_n,\varphi \left(t_n\right)\right)-f\left(t_n,y_n\right)\right|+\frac{1}{2}{h}^2\left|{\varphi }^{″}\left(t_n\right)\right|\le \alpha \left|E_n\right|+\beta h^2$, then $\left|E_n\right|\le \beta h^2\frac{\left({\alpha }^n-1\right)}{\alpha -1}$.

(c)

To determine

To show: That ${\left(1+hL\right)}^n\le e^{nhL}\text{ and }\left|E_n\right|\le \frac{e^{nhL}-1}{L}\beta h$.