Chapter 8.3, Problem 16P

Consider the initial value problem

$\begin{array}{cc}y\text{'}=3t^2/(3y^2-4),& y(0)=0.\end{array}$

a) Draw a direction field for this equation.

b) Estimate how far the solution can be extended to the right. Let $t_M$ be the right endpoint of the interval of existence of this solution. What happens at $t_M$ to prevent the solutionfrom continuing farther?

c) Use the Runge-Kutta method with various step sizes todetermine an approximate value of $t_M$.

d) If you continue the computation beyond $t_M$, you can continue to generate values of $y$. What significance, if any, dothese values have?

e) Suppose that the initial condition is changed to $y(0)=1$. Repeat parts (b) and (c) for this problem.