Chapter 8.4, Problem 6P

In each of Problems 1 through 6, determine approximate values of the solution $x=\varphi (t),y=\psi (t)$ of the given initial value problem at $t=0.2,0.4,0.6,0.8$, and $1.0$. Compare the results obtained by different methods and different step sizes.

a) Use the Euler method with $h=0.1$.

b) Use the Runge-Kutta method with $h=0.2$.

c) Use the Runge-Kutta method with $h=0.1$.

$\begin{array}{cc}x\text{'}=\exp (-x+y)-\cos x,& y\text{'}=\sin (x-3y);\end{array}$

$\begin{array}{cc}x(0)=1,& y(0)=2\end{array}$.