Homework 10, due 15 December 2005.

[goodwine]

Consider

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1. Write this second order equation as a system of two first order equations.
2. Determine all the equilibrium points for this system.
3. By computing a Taylor series approximation to the nonlinear term, determine a linear approximation to the nonlinear system near each equilibrium point. This should be in the form of one second order differential equation.
4. Solve each linear approximation near each equilibrium point.
5. Plot the solution to the nonlinear equation (determined numerically using any method/program you choose) and the solution to the linear approximation for various initial conditions on the same phase plane to verify that the linear approximation is only valid near the equilibrium point. By inspecting the graph, provide an estimate of the region in which the linear approximation is valid.
6. By computing the Jacobian, determine the linear approximation to the system of two first order equations near each equilibrium point. This should be in the form of two first order differential equations.
7. By computing the eigenvalues and eigenvectors of the Jacobian matrix evaluated at the equilibrium point, determine the solution to the linear approximation.
8. If any of the fixed points have real eigenvalues, plot the corresponding eigenvectors on the phase plane plot. Explain what they represent.