Chapter 8.3, Problem 6P

In each of Problem 1 through 6, find approximate value of the solution of the given initial value problem at $t=0.1,0.2,0.3,\text{and}0.4$. Compare the result with those obtained by the Euler method in Section 8.2 and with the exact solution (if available).

a) Use the improved Euler method with $h=0.05$.

b) Use the improved Euler method with $h=0.025$.

c) Use the improved Euler method with $h=0.0125$.

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

e) Use the Runge-Kutta method with $h=0.05$.

$\begin{array}{cc}y\text{'}=(t^2-y^2)\sin y,& y(0)=-1\end{array}$ $$