Chapter 5.3, Problem 15E

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, ap-proximate the solution to the initial value problem

$y^{″}=t^2+y^2$; $y\left(0\right)=1$, $y^{\prime }\left(0\right)=0$

at $t=1$. Starting with $h=1$, continue halving the step size until two successive approximations [of both $y\left(1\right)$ and $y^{\prime }\left(1\right)$] differ by at most $0.01$.