Chapter 8.1, Problem 19P

Consider the initial value problem

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

Where $\alpha$ is a given number.

a) Draw a direction field for the differential equation. Observe that there is a critical value of $\alpha$ in the interval $0\le \alpha \le 1$ that separatesconverging solutions from diverging ones. Call this critical value ${\alpha }_0.$

b) Use Euler’s method with $h=0.01$ to estimate ${\alpha }_0.$ Do this by restricting ${\alpha }_0$ to an interval $[a,b]$, where $b-a=0.01$.