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Unformatted text preview: Differential Equations Test 2 Solutions Answer FOUR questions. Show all working clearly. 1. A population of size N having birth rate proportional to N 2 and death rate proportional to N is harvested at a constant rate, so that d N d t = rN 2- sN- h where r, s, h are positive constants. Given that the population has size N initially, find the value h such that if h < h then N grows while if h > h then the population becomes extinct. 2. A particle of mass m falls under the influence of gravity and a resistive force proportional to the square of its speed v , so that m d v d t = mg- kv 2 where the y-axis is taken to point vertically down. (i) If the particle is initially released from rest at height h , what is its speed upon reaching the ground? (ii) What (absent ground!) is the limiting velocity of the particle? 3. Solve each of the initial value problems: (i) y 00- 4 y + 4 y = e t ; y (0) = 1 , y (0) = 0. (ii) y 00- 4 y + 4 y = 2 e 2 t ; y (0) = 0 , y (0) = 0. 4. For each of the following differential equations, write down the general solution of the corresponding homogeneous equation and the form of a par- ticular solution (without determining the undetermined coefficients):...
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- Fall '08