To determine The expression in terms of γ for the amount of salt in the tank at any time t and also, find the limiting amount of salt in the tank as t → ∞ . Explanation Definition used: The first order linear equation is of the form d y d t + p ( t ) y = q ( t ) where p and q are function of the independent variable t . Formula used: The integrating factor μ ( t ) of the first order differential equation d y d t + p ( t ) y = q ( t ) is given as μ ( t ) = e ∫ p ( t ) d t and the solution of the differential equation is given by y = 1 μ ( t ) [ ∫ μ ( s ) q ( s ) d s + c ] . Calculation: It is given that the tank contains 120 L of pure water and the concentration of salt in the mixture is γ g/L . Thus, the amount of salt initially is 0. The rate with which the mixture flows in the tank is 2 L/min , which is also the rate of the well stirred solution flowing out of the tank. Let the amount of salt in the tank at time t be Q ( t ) . Clearly, the rate of change of the concentration is given by d Q d t = rate in − rate out . For the mixture flow change when entering the tank, the rate in becomes 2 L/min × γ g/L = 2 γ g/min . The amount of dye in the tank with 120 L of a dye solution is Q ( t ) , the concentration becomes or the Q 120 . Hence, for fresh water flow change when leaving the tank, the rate out becomes 2 L/min × Q 120 = Q 60 g/min . Thus, the differential equation becomes d Q d t = 2 γ − Q 60 d Q d t + Q 60 = 2 γ such that p ( t ) = 1 60 and q ( t ) = 2 γ Compare with the formula and obtain the integrating factor of the differential equation. μ ( t ) = e ∫ 1 60 d t = e t 60 Now, obtain the solution as follows

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 2.3, Problem 2P

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The expression in terms of γ for the amount of salt in the tank at any time t and also, find the limiting amount of salt in the tank as t→∞.

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