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Solutions_Practice_Final_Math_33B_F08

Solutions_Practice_Final_Math_33B_F08 - Mathematics 33B...

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Mathematics 33B - Practice Final Instructor : D. E. Weisbart NAME (please print legibly): Your University ID Number: Your Discussion Section and TA: Signature: QUESTION VALUE SCORE 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 TOTAL 100 1
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1. (10 points) Starting at time t = 0 a solution containing 2 gm of serum per liter is being fed into a 5 liter tank at a rate of 0.1 liter per minute. The tank is full at time t = 0 and contains no serum. The mixture of liquid and serum leaves the tank at a rate of 0.1 liter per minute, so that the tank remains full at all times. It is important to stop the flow of serum when the concentration of serum in the tank (which happens to be part of someone’s body) reaches 1 gm per liter. Find the time to stop the flow. Leave your answer in logarithms. Solution: The differential equation to solve is: dx dt = 1 5 - x 50 . The solution is x ( t ) = 10+ ce - t 50 . x (0) = 0 so c = - 10. Find T so that x ( T ) = 5. T = 50 ln 2. 2
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2. (10 points) a) Find the solution to y 0 - x ( y - 2) 2 = 0 satisfying y (1) = 4. What is the interval of existence for this solution? b) Find the solution to the equation in part a) satisfying y (1) = 2. What is the interval of existence for this solution? c) Find all points in the ( x, y )-plane where the solutions in parts a) and b) cross each other. Solution: a) Separate variables and integrate to get: y ( x ) = 2 - 2 x 2 - 2 . The above function exists as a continuous function only in the intervals: ( , - 2), ( - 2 , 2), ( 2 , ). Since 1 must be in the domain of the solution, the interval of existence is ( - 2 , 2). b) y 0 is the solution. Interval of existence is ( -∞ , ). c) Solutions don’t cross. a) solution only exists in ( - 2 , 2) and in that interval x ( y - 2) 2 is continuous and furthermore ∂x ( y - 2) 2 ∂y is continuous. By uniqueness theorem, solutions can’t cross in ( - 2 , 2). 3
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3. (10 points) a) What is the general solution to y 00 - y 0 - 2 y = 0? b) What is the general solution to y 00 - 4 y 0 + 4 y = 0? c) What is the general solution to y 00 - 2 y 0 + 5 y = 0? d) To find a particular solution to y 00 - 2 y 0 + 5 y = e t + e - t by the method of undeter- mined coefficients what y ( t ) with undetermined coefficients should you use? Do not find the coefficients. e) To find a particular solution to y 00 - y 0 - 2 y = cos(3 t ) by the method of undetermined coefficients what y ( t ) with undetermined coefficients should you use? Do not find the coef- ficients. a) y ( t ) = c 1 e 2 t + c 2 e - t . b) y ( t ) = c 1 e 2 t + c 2 te 2 t . c) y ( t ) = c 1 e t cos 2 t + c 2 e t sin 2 t . d) y p ( t ) = Ae t + Be - t . e) y p ( t ) = A cos 3 t + B sin 3 t . 4
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4. (10 points) Suppose that H ( x, y ) = ax 2 + 2 bxy + cy 2 , where a , b and c are constants, and consider the system x 0 = ∂H ∂y ( x, y ) y 0 = - ∂H ∂x ( x, y ) (1) a) Show that the phase plane portraits of the solution curves of (1) are either centers or saddles, depending on the sign of ac - b 2 . Specify which sign gives centers and which sign gives saddles. Do not consider the case
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