Exam1Spring95 - 65.240 Exam #1 February 18, 1994 Your Name...

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Unformatted text preview: 65.240 Exam #1 February 18, 1994 Your Name it ()N g" r \ I\ Circle your Section Number: 1 2 3 f (I) (2 pm) (3 pm) (4 pm) Please do all 4 problems, showing your work g clearly and in reasonable detail. Give reasons for your conclusions. One 8.5 by 11 inch sheet of notes may be used. Other materials (books, papers, calculators) are not permitted during the exam. 5 Problems 1 and 2 count 26 points each, and Problems 3 and 4 counts 14 points each. The Lab question counts 20 points. § r: ll It» .0 .—- - GRADE l. (a) Suppose that y = y(x) satisfies 4y'-+- 2y = 3, y(l) = 2. (i)[q pts] Find y(x). (ii) [2 pts] What is the largest interval for which your solution is valid? < x < (iii) [2 pts] Describe the behavior of your solution as x —) co. (b) Now suppose that y = y(x) satisfies xy'+ 2y = 3, y(1) = 2. (i) [9 pts] Find y(x). (ii) [2 pts] What is the largest interval for which your solution is valid? < x < (iii) [2 pts] Describe the behavior of your solution as x —) co. 2. Suppose that y = y(x) satisfies y" + ay' + 2y = O, and a isareal number. [For example, (1 might represent the amount of damping in apendulum, if y represents the angular displacement of the pendulum bob.) (a) [7 pts] If a = 3, find the general solution for y(x). (b) [7 pts] If a = 2, find the general solution. (c) [9 pts] For one value of a between 2 and 3 (call it (to), the general solution y(x) has a different form from those above. What is do = ? What is y(x) for this case? (d) [3 pts] On the next page are three Maple plots of y' versus y and for values of 2, a0, and 3 for at. Beside each plot write the correct value for a. 3. With this winter, it's time to dream of spring! Suppose you have a 10.000 gallon swimming pool that is now half full of mildly chlorinated water [with a total of 10 oz. of chlorine dissolved in it]. In the spring you will pump strongly chlorinated water [containing 0.01 oz. of chlorine per gallon] into the pool. at a rate of 20 gallons per minute. You will also fix the pool filter to pump out the well-stirred mixture at the slower rate of 10 gallons per minute. (a) [11 pts] Write down an for the amount of chlorine c(t) in the pool at any time t. [Do NOT solve for c(t).] «t (b) [3 pts] At the right is a Maple plot of the solution to the problem from part (a). no What quantity gives the amount of chlorine in the pool when it is full 120 but not overflowing? C 100 What is this amount? 80 Joe 4. The growth of the number N(t) of a population of creatures is modeled by the so- called "Gompertz equation", which is dN a = Nina/N). m where In is the natural logarithm. (a) [4 pts] At the right is a Maple plot of the right side of (t). Find all equilibrium solutions (critical points) of (*). (b) [6 pts] Classify each equilibrium solution as asymptotically stable, semistable, or unstable. (0) There are four direction fields (labeled by A, B, C, D) from Maple shown on the next page. (i) [1 pt] Write the letter of the one corresponding to (*). dN (ii) [1 pt] Which one do you think corresponds to the equation 3 = - N [1n(2/N)]2? (iii) [2 pts] Which one corresponds to a modified Gompertz equation with a "threshold"? \o\»\'\u\~\u\~\~ \ \ \.\~\.\.\~\.\u\ \ N /////////m ///////// ///////// ///////// ///////// ///////// ///////// ///////// x/ffffff/ ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// ///////// \b\v\~\c\~\n\~\- \ \ \\\\\\\\\\\\\\\n\\\\\ z \ \\\\\\\\\\\\\\\\\\\ \\\\¢\\\\\\\\\\\n\\\\\ NNNNNNN‘ON‘O‘O‘WN‘ONN‘ON‘." NNNNN‘.\O‘INNNNN‘ONNW 0 [lfo/gl‘lu/f / / olu/f’lf-lf/ / / I’lQ/fd/fg/f -/ / T./¢Io/o/o/././ / / l A: LES: \‘\\\¢\O\\\N\\\\\\‘\\\o\o\v \\\\\\\\\\\\\\\\\u\\c\ \\\\\\\\\\\\\\\\\.\\\ \\\\N\‘\o\\.\\‘\.\.\\.\.\\\.\o )/””///////’0’0////"./’/0 //////////////’////// //’//’//////’/‘/////’//’// 65.240 LAB EXAM 1 Spring 1994 Name: ____ Sec. No. l 2 3 ** Books, notes, and Maple may be used. No discussion with other students ** Let y = y(t) satisfy the following differential equation: 91-42 _1 fl d[—e cost 3t+3y 1. (4 pts) Find the direction field for this equation on a rectangle that has t = O as one side and y = i 4 as two other sides. Write one Maple plot command that you used to produce a nice picture: NOTE: Do NOT print out a copy of your direction field, and do not bother writing any other commands. 2. (4 pts) From your picture briefly describe how solutions behave. In particular, how do solutions appear to behave as t becomes large? 3. (4 pts) Use Maple to find a formula for the general solution of your equation. Write the formula here: 4. (4 pts) Let soll be the solution that satisfies y(0) = 2 and 5012 be the one that satisfies y(0) = - 2. Use Maple to find $011 and $012 and to plot them. Then sketch (by hand) your solutions on the axes below and label them. Put scales on the axes. Y 5. (4 pts) Now find the solution that satisfies y(0) = yo. Is there a value of y0 so that the solutions behave differently, for large t, when y(O) is above or below yo? Ifso, give that y0 value. If not, explain why not. (3‘ ...
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Exam1Spring95 - 65.240 Exam #1 February 18, 1994 Your Name...

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