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Unformatted text preview: INTRODUCTION TO DIFFERENTIAL EQUATIONS, FINAL EXAM
Spring 2004 TA (circle one) Jay Polina Name Instructions. You are allowed to use two 8 1/2 X 11 inch sheets of paper of notes. No
calculators, PDAS, computers, books, or cellular phones are allowed. Do not collaborate
in any way. In order to receive credit, your answers must be clear, legible, and coherent. 1. ( 2 points each) True or False? a) T The ﬁrst step in solving ii + 412 + 4u = F(t) is to solve ii + 411 + 4n = 0. b) F The natural (angular) frequency for ii + 4u = 0 is w = 4. c) F All solutions of ii + 411 + 4U, 2 cost can be found by plugging u = A cost + B sint
into the differential equation and solving for A and B. d) LThe steady—state solution of ii + 411 + 4n = 4 is u = 1. e) T To solve it + 4cosu = 0, rewrite the equation as du/ cosu : ~4dt and integrate
both sides. f) _:__A matrix with real entries can have complex eigenvalues. g) Llf f (t) and g(t) are linearly independent solutions of ety” + y’ +y = 0, then every
solution of ety” + y’ + y = 1 can be written y(t) = af(t) + bg(t) + 1 for some constants a andb. _1W_l+réi
2 f: 1. r=1 Z 7 T
h) T Solutions of ii + it + u = 0 oscillate more slowly than solutions of ii + u = 0. linear sy In th s a dxim tes the nonlinear system
. Niko saga/EFF g 9’ E
—1 0 ' 2': = yem, g) = —:1:ey ear the on 18 t t1'1=u—lise:‘3_. j) T The integrating factor for the differential equation 6 2. A satellite photo taken at noon on May 1 shows suspicious activity near Clearwater
pond. Clearwater pond, which is known to have volume V, is fed by a fresh—water stream
that ﬂows (both upstream and downstream of the pond) at a rate F liters/ day. On the
morning of March 1, water samples taken both upstream and downstream of the pond
show that the water is clean. On the next pass of the satellite on June 1, a photo of the same area shows the pond
volume V and upstream ﬂow F unchanged, but it also shows that the nearby company
Pollution R Us has constructed a pipe that empties into the pond, and the downstream ﬂow voleme is now F + L. Water samples taken on the same day show that the upstream
water is still clean, but the water downstream now contains the toxic Compound Q. The
downstream concentration is q. To estimate the concentration p of Compound Q that the company is pumping into
the pond, assume that the water samples taken on June 1 represent steady—state values.
a) (5 points) Find a formula for p in terms of q, V, L, and F. Hint: You should set up a differential equation, but to answer the question you don’t
actually need to solve it. _ — — \' «A “ *c\¢
0H: " ‘0 by} y " ‘ a 8 I
2 \OL = 2(F+L\ => p12 FLL
L
2 (P'L) /L
p : ________
b) (15 points) Is it reasonable to assume that the system has reached steady—state in one
month? To ﬁgure this out, ﬁnd a formula for the time at which the concentration reaches
90% of its steady—state value. More space for your work is provided on the next page.
PM. P
4&1 + CANAL) : Y’L aw‘ 6110\20 *5)
0H, “? LV
 x0
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F+L PH.
ELL,E V f
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di: c, :: ,.._
F’l'L
PH.
— 4: Fat. ,
G V  ~—— 1‘ .. +L
e ‘ V
_ pL,_\/_ e, +c. Cu“: ﬂiv (‘_e€v>3
F+L FAvL
_ F+L
Q : w + c e. *9 t C°"C‘4\V'\:oa "S Gl/Vu
PH. '
3““l3 S‘Hhc VAW {a Y’LI/F‘Hx
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Law 707 EL: L (l—e, V)
t_ FJrL. FH
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" '0 :aewJ  “(nut 3. (5 points) Sketch the function f to which the following series converges. Label important
points on your axes, and extend your drawing for several periods. f(a:) = 1 + 2 bn sin mv where bn 2%. sin n3: dm
Clix/V O 4. (2 points each) Fill in the blank with the letter corresponding to the best description.
Use SHM 2 simple harmonic motion; OD = overdamped; CD = critically damped; UD
underdamped; R : resonant ; B : beating; TSS = transient plus steadystate; E
exponentially growing. a)LO_a+4a+4u=0 Manna—.0 (wraizc’ (“’7 b)§—HL71+4“=0 P1W=o =3 r: :2: c) T55 il+47i+4u=cost d) R il+4u=coth e) 0 il+4u=cos3t MM— 1)R j)——L f 5. (10 points) Solve the partial differential equation gum + uyy = 0 in the rectangle
[0,1] X [0,71'] subject to the boundary conditions u(0,y) = 0 = u(1, y), u(a:, 0) = sin 27m,
and u(x, 7r) 2 0. You do not need to Show all your work for allcases that arise; just be
sure you have found all the solutions you need. 51 £4093]: ‘1)("Y+ XYnzo 3,! :— «l’l  _l\
I!» H x X +>x :o Y — ‘1 x Y z 0
X103: xm=o Ym=o
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a w mmW=o m Y: w (amino _ A“ 63:4773 + 6’ 3n73
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I e ems: 6. ( 2 points each) For the following, ﬁnd the eigenvalues and determine Whether the
behavior near (07 0) is stable (S) or unstable eigenvalues stability
a) l _3 ¥ 513255—23; ([ '23
21:33; 0 3
y=—3y 0 '3
+r J—° olul’ —r 3 z
c) 4' 6' —~S x233; (0 3» ’zr
yz—Zfl; "'2 0
(1'4 C :0
d) r, ’3 _5 abz—x—3y [A '3’)
y=—3y O '3
e) I '—‘ _u ¢2m+2y+y2 (l 7.)
yz—y+:c2 0 ”\ 8. (5 points) Solve the initial value problem 11 = u — 2v, '1') = 32), u(0) : 1, v(0) = 2. (l ’2, efaenvxyu an. \) 3 _3__ (—2 Z>(j‘ a
O 3 l o 1 6 0 ° 7‘) 0) ...
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 Spring '04
 Yoon
 coherent, Partial differential equation, ety, pond. Clearwater pond, Jay Polina

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