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Unformatted text preview: Alex Boisvert Math 33B, Winter ’08 February 1, 2008, 8:00 AM Midterm 1
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Check your section: _ 2a _ 21) (Th) TA: Neel Tiruviluamala
_ 2c _ 2d (T11) TA: Eric Radke This is a closedbook exam. Do not use notes, books, papers, or electronic devices of
any kind. Do all work on the sheets provided. Do not use your own paper or blue books.
If you need more space for your solution, use the back of each page; you may request
extra paper. Be sure to state clearly if you are continuing on a different page and label
the problems well. Do all 5 problems. For full credit, you must show all your work. Do not worry about
oversimplifying your answers. Please clearly indicate your ﬁnal answer, for example by putting a box around it. Problem Out of Points (1) ( points) Radioactive substances decay according to the differential equation
N’ = —AN where N (t) is the amount of the substance remaining at time t and A is a.
constant1 called the decay constant. (a) Solve the above differential equation to ﬁnd N(t). ﬁ: —«N\) “:3 319 ~gﬂkd‘trb lhlmi‘ fiAHC l .
(b) Show that after a period of T; = X, the material has decreased to 6—1 of its
original value. T; is called the time constant. N CW = A e?” y" = e?"
“('0le “Shea A64 '2 A08”) «HM; Fs’l‘rvﬁ— (C) A certain radioactive substance has halflife of 10 hours. Compute its time constant.
,_1 "10} \ F; 015‘
mg?” ' e i? "—326 \
9/?— ==*\> —\n[2\:“'0> =~> >6: ‘55) [’0
l
TX; VA: X101) (2) ( points) Only one of the following diﬁerential equations is linear. Determine
which one it is and solve it using any method from class. (a) ’=y+y2‘ (b) y,+ 93 = cost
if t2 (c) y’—2ty = cosy (d) y’=t/y (3) ( points) A tank initially contains 100 gallons of pure water. Water begins
entering the tank through two pipes: through pipe A at 6 gal/min and through
pipe B at 4 gal/min. Simultaneously, a drain is opened at the bottom of the
tank1 letting solution ﬂow out at 10 gal/ min. (a) Supervisors quickly discover that the water coming through pipe B is
contaminated, containing 0.5 lb of pollutant per gallon. If this process runs
for 10 minutes1 how many pounds of pollutant are in the tank after this
10minute period? (Call this number sup for use in part (b) of this problem) M as: itsel atats as t.
xii—,(mk M “@lri as): 9 togb L Qoﬂx __ 313; H at
X — To— ‘ zo'x “’ "TB"
«is , “Jr/:0 r' lo
:> ﬁlm—xi: We am: 'xllr\=QO—Aa { 170 =~>A=30
[X 0 =3 will? {10*306 m We, U90} ’Kllol rm (b) The supervisors shut off pipe B after this 10—minute period, allowing pipe A
and the drain to function as before. How much pollutant remains after 10
minutes of this new process? (Use the notation 3,, instead of the potentially
complicated number from part (a)) (no: at “05 Witt at “inhale r .2:— it a
(K : ﬁlo ‘23, *FO'Oo_q€ :3 SQ, (ucimﬁhciuTLHU
X ﬂ POD—Lat '25 \“M‘: "3‘3 inhooLrtl 4c, 10. .
=‘~> «smiooewﬁ' 10:09:: WW 1—10 «(oi : M‘OOSVHC«p A: eye)?
“)9 e xiii: «icy105‘“: (1004M iolq we Wt mm c mp, too"°"’(to') . (4) (points) Solve the following differential equation. Hint: Check for exactness, homogeneity. If neither holds, assume that the integrating factor is a function of
either :1: alone or 3; alone. (59+ y)siuy do: + (xsiny+ cosy) dy = 0 1:; PciX +6443
P
~31; 2 (#330393 +80% n, 3%: owing 330 “BA? emcy“
Nag mi th3‘ _,_,La£’.,m @2921 a.
SNCQCéup(B‘J a): _ 2 Cori : é‘ManZH SHHHW. wewmrwé W chiWQ“ Ciﬂiazor LAWN
a 29ng ciaan to oz we}.
Rm): $P4x + (9(3) 2) Rag): 'L’xzir gx “9(5). (PM: iSPier= VG” SW " 3“"? (5) ( points) Consider the initial value problem 33’ 2 —:L'2cost, =1 (e) Use the existence and uniqueness theorems to prove that. a solution to this
differential equation exists and is unique. rxi; ﬁxra: F’XQCDgL‘ £(x,E\ 1‘3 0A (b) Solve this differential equation and give the interval of existence of the
solution. ...
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