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
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
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

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## This note was uploaded on 02/08/2010 for the course MATH 33B taught by Professor Staff during the Winter '07 term at UCLA.

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Solutions_Practice_Final_Math_33B_F08 - Mathematics 33B -...

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