Chapter 13.2, Problem 15E

To determine

(a)

To show:

The problem

$\begin{array}{cc}{x}^{\prime }\left(t\right)=-y^2\left(t\right),& x(0)=0;\\ y^{\prime }\left(t\right)=z\left(t\right),& y(0)=1;\\ z^{\prime }\left(t\right)=x\left(t\right)y\left(t\right),& z(0)=1\end{array}$

is equivalent to the system of integral equations

$x\left(t\right)=-{\displaystyle \underset{0}{\overset{t}{\int }}{y}^2\left(s\right)ds}$,

$y\left(t\right)=1+{\displaystyle \underset{0}{\overset{t}{\int }}z\left(s\right)ds}$,

$z\left(t\right)=1+{\displaystyle \underset{0}{\overset{t}{\int }}x\left(s\right)y\left(s\right)ds}$.

To determine

(b)

To find:

The iterations $x_1\left(t\right)$, $y_1\left(t\right)$, $z_1\left(t\right)$ and $x_2\left(t\right)$, $y_2\left(t\right)$, $z_2\left(t\right)$.