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Unformatted text preview: 65.240 Exam #1 February 18, 1994 Your Name it
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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 wellstirred 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\~\~ \ \
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//’//’//////’/‘/////’//’// 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: 9142 _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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 Spring '04
 Yoon
 Differential Equations, Equations, Chlorination, pts, Stability theory

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