Chapter 6.6, Problem 1CP

Apply Backward Euler, using Newton’s Method as a solver, for the initial value problems. Which of the equilibrium solutions are approached by the approximate solution? Apply Euler’s Method. For what approximate range of h can Euler be used successfully to converge to the equilibrium? Plot approximate solutions given by Backward Euler, and by Euler with an excessive step size.

$\begin{array}{l}\text{a}\text{. }\{\begin{array}{c}y\text{'}=y^2-y^3\\ y(0)=1/2\\ t\text{ in}\left[0,20\right]\end{array}\\ \text{b}\text{. }\{\begin{array}{c}y\text{'}=6y-6y^2\\ y(0)=1/2\\ t\text{ in}\left[0,20\right]\end{array}\end{array}$