First Order Equations- Solution

First Order Equations- Solution - GE 207K 1 Problems 1.1...

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Unformatted text preview: GE 207K 1 Problems 1.1 Problem 1 Consider a tank that contains 5 gal of pure water at time t = 0. Assume that water containing lb/gal of sugar enters the tank at a rate of r gal/min, and that the well-stirred mixture of sugar and water drains from the tank at the same rate of r gal/min. First we want to find an expression in terms of and r for the amount of sugar in the tank at any time t . Let Q ( t ) denote the amount of sugar in the tank at time t , where Q is expressed in pounds and t is expressed in minutes. The rate of change in the amount of sugar in the tank can be written as dQ dt = r- rQ 5 which can be rewritten as rQ 5- r + dQ dt = 0 . We can find an integrating factor to turn this into an exact equation; in particular, we choose ( x ) = e rt/ 5 . This yields rQe rt/ 5 5- e rt/ 5 r + e rt/ 5 dQ dt = 0 and so we obtain the following expression for the amount of sugar in the tank Q ( t ) = 5 + Ke- rt/ 5 where K is a constant. Finally, we use the initial condition Q (0) = 0 to solve for K and obtain Q ( t ) = 5 - 5 e- rt/ 5 . Next we want to find an expression for the limiting amount of sugar in the tank as t approaches infinity. Since e- rt/ 5 approaches zero as t approaches infinity, the limiting amount of sugar in the tank is Q L = 5 . 1 1.2 Problem 2 Assume that a sum S is invested at an annual rate of return r that is compounded continuously. First, we want to compute the time T that is needed for the original investment to quadruple in value. Let S ( t ) be the value of the investment at time t , where t is expressed in years. Since interest is continuously compounded, the rate of change of the value of the investment can be written as dS dt = rS. Solving this equation yields S ( t ) = S e rt (1) where we have applied the initial condition S (0) = S . We want to find T such that S ( T ) = 4 S , which is equivalent to finding T such that e rT = 4; this yields T = ln(4) r . Next, we want to compute T assuming that r = 5%. Plugging this into the above expression, we obtain T 27 . 7 years . Note that r has the units of years- 1 . Finally, we want to compute the value of r that is needed for the original investment to quadruple in 11 years. This is equivalent to finding r such that e 11 r = 4; this yields r 12 . 6% . 1.3 Problem 3 A ball with mass 0.08 kg is thrown upward with initial velocity 25 m/s from the roof of a building 10 m tall. You can neglect the effects of air resistance in this problem. Assume that the radius of the earth R = 6380 km and that the acceleration due to gravity g = 9 . 8 m/s 2 ....
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This note was uploaded on 10/08/2010 for the course GE 207K taught by Professor None during the Fall '10 term at University of Texas at Austin.

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First Order Equations- Solution - GE 207K 1 Problems 1.1...

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