math_2270test_091s - f/ufl‘OYLW Instructor:H Zhu April 1...

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Unformatted text preview: , f/ufl‘OYLW Instructor:H. Zhu April 1, 2009 MATH 2270: Differential Equations TEST I Last Name: __—First NaIne:*_ ID: 1. E Find the general solution in vector form and solve the initial value problem: 2m + 3ty PM [JULLAj lbw/Old 2ty 110(0) 2 1, y(0) = 1. . Ha (e\ 8\ H II .30 By changing variable(s), write the following equations into either separable or linear equations. If it is a linear equation, find one integrating factor. You do NOT need to find the general solution/\_,.—-—— (mfi— 562+? —, / kit-Yr: W— $2_y2- \/ H505 b) COS’LUd—w = t Sinw+ 1 1-H. ( I d 1+t2 14-752. (gfiz y . d2: $+y3 1 fl / / 417 du :Z “PU ‘TX W A ”’f M U 2cm +u:/ H}; it}. _. 0“ 2 O“ “ +153? “U +\ 3. IE Consider the fish population model with harvesting: dP 1 P 8 5:31)“??- (a) Find the equilibrium solutions of the model. (b) Classify each of these equilibrium solutions as sink or source. (c) Plot the phase line. (d) Find the general solution. (e) Discussion: given the initial fish population size P(0) = P0 > 0. What do these results imply about the long term behavior of the fish population? Answer this question for the case when P0 = 1, P0 = 6 and P0 = 12 and plot the solutions. (f) Consider the model with harvesting dP 1 P —=— 1———h dt 3“ 10) where h > 0 is a parameter. Locate the bifurcation value for the one—parameter family and draw the phase lines of parameter slightly smaller than, slightly larger then, and at the bifurcation value. @- hf ELK/49460 fzp ...
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