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Unformatted text preview: 65.240 LAB ICXAM I Name Sec. 20 21 10 ll 12
*** Books, notes, Maple, etc. may be used. No discussion with other students.*“
let y = y(t) satisfy the following differential equation: ‘12 7 3
dt t + exp( ..t)cost + 2y 1. Find the direction field for this equation on some rectangle that has I = 0 as one side
and that includes y =0. Write the 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. From your picture briefly describe how solutions behave. In particular, how do solutions
appear to behave as t becomes large? 3. Use Maple to find a fomiula for the general solution of your equation. Write your formula here: 4. Let sol 1 be the solution that satisfies y(()) = 3 and sol 2 be the one that satisfies y(0) = — 3 . Use Maple to find sol 1 and so] 2 and plot them. Then sketch (by hand)
your solutions on the axes below and label them. Put scales on the axes. y 5. Now find the solution that satisﬁes y(0) = yo. I: there a value of y0 so that the
solutions behave differently, for large t, when y(()) is above or below yo? If so, give
that y0 value. If not, explain why not. (33.240001, (J3. (Jo 1e51, l Scl'yteuﬁml' ‘21), 13‘ (3 NAME: _V __ , e V Aw”, SE'f’T'l‘ION 01 05 ()6
Sl‘)1\'l‘_H l’lmsv nnsvwl’ (rm‘h of the following questions. Show your Work in detail ulnl give reasons for your answers whenewr appropriate. You may refer to one sheet of notes, but no other books, references, calculators etc. are permitted. PROBLEM # POINTS Tt ) t a 1 (HM ma (35.210001. U3. (J0 .\.\.\lla: 3. (A) Find l.l(}h’(1llll,lt)ll ul lllL‘ initile \‘nluv proliluni (13/ cos .1? ‘17—— l 31 I I , ('v—H' 2 ::
(LL' T J 1'2 J“ / l ()1! What, interval is the solution valid? (ll) [“inil llw general solution ul llio (lillvi'vnlinl vqnnliun 21/" + 5y' + By : (l. 0.7.2100 u l‘ u.) m; ILLIHuW“ , _ . l. (a) l"ind the explirit solution y 2: (0(4) ot the initinl value problem (15/ c" — 2.17 ‘ ~———, vé)‘ ~I.
(1.1: '2] M i (h) Consider the equation
41y _ m — y(u  llty — 5). (i) l’ind the (‘qllllllJl‘llllll solutions ((‘i‘iliczil poiuls).
(ii) 011 the axes helow draw a rough sketch of the direction held for 0 S t S 10, ~13 :1 y S 8. (iii) For each equilibrium solution1 determine whet her it is an attractor (stable), or a repeller
(unstable). (iv) Sketch the graph of the solution that sntishes the initial condition y(0) : 3. What limit
(if any) does this solution approach as t «a 00'? ‘3 l (M3.JlUUUl, n7“ Um Nﬁ‘ll‘lz 7 5. (a) l“in«l the (‘lgt‘HVulllt‘n ninl ('igt‘nvm‘tul‘s ol‘ the matrix 'l‘hen find the general solution of the system (h) The population of a certain colony of ants grows at a rate that Causes the population to douth
each month (if the colony is left undisturbed). An anteater limls this colony and eats 10,000 ants per
day. lnitially there are 350,000 ants in the colony. Formulate an initial value problem describing the
way in which the ant population changes. Be sure to state the units for the independent and dependent variables. DO NOT SOLVE the problem you formulate. 65.240 Exam #1 Your Name __ Circle your Section Number: 10 ll 12
(2 pm) (3 pm) (4 pm) Please do all 4 problems, showing your work clearly and
in reasonable detail. Give reasons for your conclusions. One 8.5 by ll inch sheet of notes may be used. Other
materials (books, papers, calculators) are not permitted
during the exam. Problems 1 and 2 count 32 points each, and Problems
3 and 4 count [8 points each. 20
(9 am) (10 am) October I, .993 21 

I
V l
4 :
1 i
r w l. (_u)[12pts] Find the solution of ((11% — 3y = 6, y(()) = 1 (b) Ill) pts] Find the general solution of %% — 3y = 6 — 4cm. (*) (c) l 4 pts] Find the solutions of (*) with the initial condition y(()) r: y“. (d) [ 2 pts] What is the largest interval for which your solution in pan (0) is valid? (e) [ 4 pts] Describe the behaviour of your solution in part (c) as t —) 00. 2. (2t) Suppose y(x) satisﬁes y" + Zy' ~ 8y : 0. (i) [10 pts] Find the general solution (containing two arbitrary constants) for y(x). (ii) [6pts] Find the solution satisfying y(()) = l, y'(()) = —3. (b) The next two questions are related. 1 4
(i) [ 8 ptsl Find the eigenvalues and eigenvectors of A =[1 ’7] . 1.. (ii) I 8 pts] Find the general solution (containing two arbitrary constants) of the system
dxl/dt = x1 + 4xq, de/dt = x1 — 2x,. 3. Suppth that two water tanks, each of volume 50 gallons, are interconnected.
lnitially, tank l is lull of fresh water and tank 2 is full of water containing 25 ounces
of salt. Water containing 3 ounces of salt per gallon ﬂows into tank 1 at a rate of 2
gal/min. Tank 1 is thoroughly stirred, and the mixture is removed in two ways: 3
gal/min flows into tank 2, and l gal/min is drained away from the system. Tank 2 is also well stirred. Its mixture flows into tank 1 at a rate of 2 gal/min,
and l gal/min is drained away from the system. (a) [12 pts] If Q10) and Q20) are the amounts of salt in tanks 1 and 2 at time t ,
write differential equations and initial conditions satisfied by Q1 and Q2. (b) [ 6 pts] Without solving the differential equations, find the equilibrium values (that
is, having no change with time) of Q1 and Q2 for this system. 4. A second—order chemical reaction, involving two molecules of the same substance
that interact to form one molecule of a new substance. is governed by the following
equation: 2
a = (P * X) (*)
where x(t) is the concentration of the new substance at time t and p is a constant. (a) [4 pts] Find all equilibrium solutions (or critical points) of (*). (b) [8pts] Without solving (*), sketch dx/dt versus x. (c) [6pts] What is the concentration of x(t) in the limitof large t, ifthe initial
concentration x(O) is (i) x(0) = P?
(ii) 0 < x(O) < p?
(iii) x(0) = 0? LABORATORY TEST 1A
65.2400—04,05,00 September 28, 1993 NAME _____________________________ _
SECTION _________ _
SEAT ______________ _ 1. Consider the differential equation dy __ t 2y 1......— cit—23 (a) Draw a direction ﬁeld for t _>_ 0 and a suitable y range. Your direction ﬁeld should
suggest to you that solutions of the given equation are approaching a certain asymptote as t becomes large. From the direction field on your monitor estimate the slope of the asymptote. Ans ___________________________ __ Estimate an equation for the asymptote. Ans _________________________ _ (b) Draw a plot that includes the solution passing through the point (0,1). Estimate the value oft for which this solution crosses the t axis. Ans __________________________ _ (c) There is one solution of the given equation whose graph touches, but does not cross, the t axis. Estimate the initial value for this solution, that is, its value when t := 0. r Ans _________________________  ‘l 2. Consider the initial value problem (a) Use dsolve to find the solution of this initial value problem. Write the solution in the space below. (b) Plot the solution for 0 g t S 10. On the axes below sketch the graph shown on your monitor. Be sure to indicate the scale on each axis. 5 f
i . _ u,__v~_._i._r__ _ _7 .M..__._._» t (c) Determine the coordinates (to five decimal places) of the first local maximum point to the right oft = 0. Ans _____________________________ . (d) Determine the value (to live decimal places) of y when t = 5. Ans _________________________ _ ...
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 Spring '04
 Yoon
 Differential Equations, Equations, Constant of integration, Boundary value problem, general solution, sol 1

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