Key Concepts center characteristic equation complete solutions direct method eigenvalues eigenvectors general form global stability homogeneous solutions improper stable node improper unstable node isocline isoscctors local stability particular solutions phase diagram phase plane saddle path saddle point saddle-point equilibrium simultaneous system stable focus stable node steady state substitution method trajectory unstable focus unstable node

Review Questions

1. Explain how the characteristic equation can be derived and how it is used to find the characteristic roots for a. differential or difference equation.

2. Under what conditions is the particular solution given by the steady-state solution for (a) a system of two differential equations, and lb) a system of two difference equations?

3. If the steady stale of a system of two differential equations is a stable focus, sketch the paths that y\ and y2 would follow as a function of time.

4. Why is a saddle-point steady stale said to be unstable even though the saddle path converges to the steady state?

5. State the conditions under which a system of differential equations is stable.

6. State the conditions under which a system of difference equations is stable.

Review Exercises l. Solve each of the following systems of linear differential equations using the substitution method and determine the stability property of the steady state:

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