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Unformatted text preview: Math. 224. Review problems 2 Spring 02 David Gurarie Name Topics ² Linear models : oscillators, sliding chain, competition-cooperation, migration, buyer- seller markets; boundary value problems (equilibrium heat distribution) ² Fundamental pair, general solution, IVP-solution, BVP-solutions ² Phase-plane, equilibria ² Eigenvalue method: real, complex, repeated ² Applications (damped oscillators, migration et al.) ² Bifurcations (trace-det plane) ² Method of characteristic polynomial for 2nd (and higher) order DE’s ² Forced oscillations: characteristic potential and undetermined coe¢cients ² Laplace transform method Sample problems 1. Modeling (a) Competition-cooperation: show that competing/cooperating species f x;y g with the natural growth rates a;b , and interaction coe¢cients f b;c g obey a di¤er- ential system with matrix A = · a § b § c d ¸ : (i) Explain the meaning of § sign in terms of competition-cooperation (ii) Show that competition/ cooperation matrix A has real eigenvalues. (iii) What happens to eigenvalues of A when b has positive sign and c negative ? (b) Write DE, DS models for sliding chain (c) Predator-prey: write linear di¤erential model for a predator-prey model, and discuss its eigenvalues and behavior of solutions....
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This note was uploaded on 05/09/2008 for the course MATH 224 taught by Professor Hahn during the Spring '07 term at Case Western.
- Spring '07