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Unformatted text preview: The University of Sydney Math1003 Integral Calculus and Modelling Semester 2 Exercises for Week 11 2011 Assumed Knowledge Integration techniques. Objectives (10a) To be able to solve differential equations that are separable, linear or both. (10b ) To be able to construct and solve equations describing flow and mixing problems. Preparatory Questions 1. Classify each of the following differential equations: ( i ) (1 + x 2 ) dy dx = 1 2 xy ( ii ) (1 + x 2 ) dy dx = y 2 xy Practice Questions 2. Find the general solutions of the differential equations in Preparatory Question 1. 3. In a prolific breed of rabbits the rate of birth and the rate of death are each proportional to the square of the population N . Let t be a time variable and k 1 ,k 2 be the constants of proportionality for the rates of births and deaths respectively. Assume k 1 > k 2 and write k = k 1 k 2 . Then dN/dt = kN 2 , where k > 0. Solve this differential equation to show that N ( t ) = N (0) 1 kN (0) t , where N (0) is the initial population. Suppose we start with an initial population of 2 rabbits and that there are 4 rabbits after 3 months. What does this model predict will happen after another 3 months? 4. (Suitable for group work and discussion.) When people smoke, carbon monoxide is released into the air. In a room of volume 50 m 3 , smokers introduce air containing . 05 mg / m 3 of carbon monoxide at the rate of 0 . 002 m 3 / min. Assume that the smoky air mixes immediately with the rest of the air, and that the mixture is pumped through an air purifier at a rate of 0 . 002 m 3 / min. The purifier removes all the carbon monoxidemin....
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This note was uploaded on 09/01/2011 for the course YEAR 1 taught by Professor Various during the Three '11 term at University of Sydney.
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