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Unformatted text preview: Math 42 — Chapter 7 Practice Problems — Set A 1. Show that y = ln t + 3 t is the solution of the initial value problem: t 2 y + ty = 1 , y (1) = 3 . 2. Show that y = 1 √ 5- t 2 is the solution of the initial value problem: y = ty 3 , y (1) = 1 2 . 3. Find the solution of the initial value problem: y = e- y 2 ( t + 1) yt 2 , y (1) = 2 . 4. Find the solution of the initial value problem: y = te- t y 2 , y (0) = 2 . 5. In a certain country the population grows according to natural growth with relative growth rate k = 1 10 per year, but crowding also encourages a certain number of people to leave. Suppose that people are leaving the country at a rate 1 10 √ P million people/year, where P is the population in millions. (a) Write a differential equation which models the growth of the population. (b) What are the equilibrium values of the population? (c) What will happen in the long term if there are initially half a million, one million, or four million people? (d) Suppose there are initially P = 4 million people. Use Euler’s method with h = 5 to estimate what the population will be in 10 years. (e) Use the differential equation to find an exact expression for the population after t years if the initial population is 4 million. 6. A certain population of animals is dependent on a seasonally varying food supply. The rate at which the population grows is proportional to both the current population size P and to cos( π 6 t ); i.e., it is proportional to their product. (Here t is the time measured in months.) Suppose the initial relative growth rate (i.e., 1 P P when t = 0) is 1 6 per month, and the initial population is 1000. (a) Write a differential equation which models the growth of this population. (b) Suppose the initial population is 1000. Use Euler’s method with h = 3 to estimate the population after 12 months....
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