Solutions_Practice_MT_1_Math_33B_F08

Solutions_Practice_MT_1_Math_33B_F08 - Mathematics 33B -...

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Unformatted text preview: Mathematics 33B - Practice Midterm 1 Official exam to be administered Oct. 24th, 11:00am 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 TOTAL 50 1 1. (10 points) Determine which equilibrium solutions to the equation dy dt = y 3 (1+ y ) 2 (2- y ) are asymptotically stable and which are unstable. Then make a rough sketch showing the behavior of the solutions to this equation. Solution. The equilibrium solutions here are y ≡ - 1 , y ≡ 0, and y ≡ 2. We have dy dx ( x ) < for x > 2, dy dx ( x ) > 0 for 0 < x < 2, dy dx ( x ) < 0 for- 1 < x < 0, and dy dx ( x ) < 0 for x <- 1. Thus, y ≡ 2 asymptotically stable, and y ≡ 0 and y ≡ - 1 unstable. [Cannot do sketches here, but they are important.] 2 2. (10 points) Suppose that P ( t ) is the population (in millions) at time t (in years) of species that is being harvested at a rate of h million per year. If the growth rate of the population without harvesting is r , then P ( t ) satisfies dP dt = rP- h. Suppose that r = 0 . 4 and h = 0 . 5. If the population at time t = 0 is 1 (million), at what time will the population reach zero? Leave your answer in logarithms. Solution. Since this problem gives you the equation, just start with dP dt = (0 . 4) P- (0 . 5) and this implies P ( t ) = 5 4 + Ce (0 . 4) t . Since P (0) = 1, this gives P ( t ) = 5 / 4- (1 / 4) exp( . 4 t ). So solving for P ( T ) = 0 gets you e (0 . 4) T = 5 and T = 5 2 ln 5 . An alternate way to do this is to treat dP dt = rP- h. as a separable equation. That gives Z dP P- h/r = Z rdt and ln | P- h/r | = rt + C. substituting the values for h and r , and using P (0) = 1 to determine C , you get ln | P- 5 / 4 | = 0 . 4 t + ln 1 / 4 . So if P ( T ) = 0, then T = (1 / . 4)(ln(5 / 4)- ln(1 / 4)) = (5 / 2) ln 5 . 3 3. (10 points) (a) Find the solution to dy dx = 1 x (2 y- 1) that satisfies y (1) = 2, AND find the domain of existence for this solution. Solution. Since this equation has no equilibrium solutions, you can begin with Z (2 y- 1) dy = Z 1 x dx and y 2- y = ln | x | + C. Substituting y (1) = 2, you have 4- 2 = ln 1 + C . So C = 2, and y 2- y- ln | x | - 2 = 0 and y = 1 ± p 1 + 4 ln | x | + 8 2 You need the + sign to get y (1) = 2. So y ( x ) = 1 + p 9 + 4 ln | x | 2 Since negative numbers have imaginary square roots, you need 9 + 4 ln | x | ≥ 0. So we need e- 9 / 4 ≤ | x | < ∞ . However, when the solution is defined on two intervals, you have to choose the one containing the initial value as the domain of existence. So the domain of existence is e- 9 / 4 < x < ∞ and y ( x ) = (1 + √ 9 + 4 ln x ) / 2. b) What happens at the endpoint(s) of the domain of existence? Why doesn’t the existence theorem say that you should be able to continue the solution beyond them?...
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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_MT_1_Math_33B_F08 - Mathematics 33B -...

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