review-f1 - Math 224 Review David Gurarie Fall 01 1 Sample...

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Ma th .224 :R ev iew David Gurarie Fall 01 1S a m p l e p r o b l e m s 1. Find equilibria, determine stability, draw phase-line of logistic-type model: y 0 = P ( y ) -polynomial of y ,e . g . P = y (1 y/N ) ... . Write analytic solution of the logistic equation: y 0 = ay (1 y/N ) . 2. Determine bifurcation values { α } and the type of bifurcation for ODE: y 0 = P ( y, α ) .g . P = y (1 y/N )+ α . 3. Identify the following slope- f elds with 1-st order DE’s: (a) y 0 = y (1 y 2 ) (b) y 0 =co s ( y + t ) (c) y 0 = y 2 t (d) y 0 = y (1 y a cos t (e) y 0 = y +cos t . (a) Identify the autonomous equation, f nd its equilibria and sketch phase- line. (b) Sketch solution-curves for all equations. (c) Determine which of the f ve could be solved analytically, and describe the method. Give the general solution -2 0 2 4 6 8 10 -4 -3 -2 -1 0 1 2 3 4 t -2 0 2 4 6 8 10 -5 -4 -3 -2 -1 0 1 2 3 4 5 0 5 10 15 20 -1 -0.5 0 0.5 1 1.5 2 t 1
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4. Set DE models for the following systems. Describe them as DE/DS; 1-st (or higher) order; linear/nonlinear; separable, autonomous, conservative, etc. Outline solution method for each. (a) Growth with migration: two populations ( x, y ) in regions A,B grow logistically and migrate between two regions, so that fraction a of x - population migrates to B , while fraction b of y -populations migrates to A . (b) Logistic model with harvesting (discuss solution methods for constant and periodic harvesting) (c) Competing/cooperating species with logistic growth rates for each one. Explain signs ± of interaction coe cients. (d) Sliding chain with friction coe cient proportional to the horizontal (table) portion of the chain. Write it as a 2-nd order DE and the dif- ferential system. Indicate solution method with or without friction. x l-x F g =gx (e) Food chain system made of trees, moose, wolves, where p percent of moose are hunted annually (f) Tank of 100 gal half- f lled with lead-contaminated solution of 5 mg/gal is F ushed with the fresh water at the rate 3 gal/sec and the mixture is taken out at the rate 1 gal/sec. Find how long it would take to
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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.

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review-f1 - Math 224 Review David Gurarie Fall 01 1 Sample...

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