Chapter 5.3, Problem 17E

In Problems 14-24, you will need a computer and a programmed version of the vectorized classical fourth-order Runge-Kutta algorithm. (At the instructor’s discretion, other algorithms may be used.)

Appendix G describes various websites and commercial software that sketch direction fields and automate most of the differential equation algorithms discussed in this book.

Using the vectorized Runge-Kutta algorithm, approximate the solution to the initial value problem

$\frac{du}{dx}=3u-4\upsilon$; $u\left(0\right)=1$,

$\frac{d\upsilon }{dx}=2u-3\upsilon$; $\upsilon \left(0\right)=1$

at $x=1$. Starting with $h=1$, continue halving the step size until two successive approximations of $u\left(1\right)$ and $\upsilon \left(1\right)$ differ by at most $0.001$.