Chapter 8.3, Problem 15P

Consider the initial value problem

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

a) Draw a direction field for this equation.

b) Use the Runge-Kutta method to find approximate values of the solution at $t=0.8,0.9,$ and $0.95$. Choose a smallenough step size so that you believe your results are accurate to at least four digits.

c) Try to extend the calculations in part (b) to obtain an accurate approximation to the solution at $t=1$. If you encounterdifficulties in doing this, explain why you think this happens. The direction field in part (a) may be helpful