Chapter 8.4, Problem 7P

Consider the example problem $\begin{array}{cc}x\text{'}=-x+4y,& y\text{'}=x-y\end{array}$ with the initial conditions $x(0)=2$ and $y(0)=-0.5$. Use the Runge-Kutta method to find approximate values of thesolution of this problem on the interval $0\le t\le 1$. Start with $h=0.2$ and then repeat the calculation with step sizes $h=0.1,0.05,\mathrm{...},$ each half as long as in the preceding case. Continue the process until the first six digits of the solution at $t=1$ are unchanged for successive step sizes. Determine whether these digits are accurate by comparing them with the exact solution given in Eqs. (11) in the text.