Reading: Chapter 13, sections 1 and 2 only.

Exercise: Consider the equation given in number 2 of Exercise 13.2. For that equation

- Find the equilibrium points.
- By using a Taylor series expansion for the nonlinear term(s), determine a linear differential equation that approximates the nonlinear equation near
*each*equilibrium point. Find the solution for each linear equation. Which ones are stable and which are unstable? - Write a C or FORTRAN program that uses 4th order Runge-Kutta to solve the nonlinear equation.
- Near each equilibrium point, in the
*phase plane*plot the numerical solution to the nonlinear equation for several initial conditions. On the same graph, plot the solution to the linear approximation. For the linear equation, you can either use the solution you determined for the previous part, or you could find a numerical solution. Verify that as the solution moves farther away from the equilibrium point, the solution to the linear equation becomes a worse approximation to the nonlinear solution. Basically, make plots like Figures 13.8-13.13.