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Unformatted text preview: Math 3F Exam #1 Sample Laney College, Fall 2010 Fred Bourgoin 1. Solve the initial-value problems. (a) dy cos x = (1- y ) dx, y ( ) = 2 (b) y prime = 2 xy 2 + 3 x 2 y 2 , y (1) =- 1 2. Brine containing 0.2 kg of salt per liter runs into a tank initially filled with 500 L of water containing 5 kg of salt. The brine enters the tank at a rate of 5 L/min. The mixture is continually stirred and flows out at a rate of 5 L/min. (a) Find the amount of salt in the tank after 10 minutes. (b) After 10 minutes, the tank develops a leak and an additional liter per minute flows out of the tank. How much salt is in the tank 20 minutes after the leak develops? 3. Is y = 3sin 2 x + e- x a solution to the differential equation y primeprime + 4 y = 5 e- x ? 4. Recall that Newtons law of cooling states that the rate at which the temperature of an object changes is proportional to the difference between the temperature of the object and the temperature of its surroundings. A body is discovered at noon on a cold Southern California day (16 C); the temperature of the body is then recorded as 34.5 C. When the homicide detectives show up an hour later, the bodys temperature is 33.7 C. When did the murder occur? (The normal temperature of a living human being is 37 C.) 5. Without solving the problem, determine the largest interval in which the solution to the initial-value problem is certain to exist. (4- t 2 ) y prime + 2 ty = 3 t , y (- 1) =- 3 6. For each differential equation, determine its order and decide whether it is linear or nonlinear. (a) d 2 y dt 2 + sin( t + y ) = sin t (b) d 3 y dt 3 + t dy dt + (cos 2 t ) y = t 3 7. Solve the linear equation. (12 pts) y prime + (cos x ) y = cos x 1 8. Consider the differential equation dy dt = ( y + 2)( y- 2) 2 . (a) What are the equilibrium solutions? (b) Draw a phase line. (c) For each equilibrium solution, determine its stability. (d) Sketch the equilibrium solutions a few other solutions....
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