Chapter 2.7, Problem 20P

(a)

To determine

To show: The exact solution of the initial value problem $y^{\prime }=1-t+y,y\left(t_0\right)=y_0$ is $y=\varphi \left(t\right)=\left(y_0-t_0\right)e^{t-t_0}+t$.

(b)

To determine

To show: The equation $y_k=\left(1+h\right)y_{k-1}+h-h{t}_{k-1},k=1,2,\dots$ by using Euler formula.

(c)

To determine

To show: The equation $y_n={\left(1+h\right)}^n\left(y_0-t_0\right)+t_n$ by method of induction provided $y_1=\left(1+h\right)\left(y_0-t_0\right)+t_1$.

(d)

To determine

To show: The value of $y_n\to \varphi \left(t\right)\text{ as }n\to \infty$, by choosing $h=\frac{t-t_0}{n}$.