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**Unformatted text preview: **Math 54 — Sample Final Exam
August 15, 2008, 08:00—10:00 Name: \v i on This is a closed book, closed notes exam. Calculators are not allowed. You
have two hours to complete the exam. To receive full credit, write legibly, show
your work and write proofs in complete sentences. If you need more space, use
the back of the page of the problem on which you are working. —-— 1. Find the solution of the initial—value problem 2y” + 99, + 10y = 8sint + QCost
31(0) = 4, y’(0) = —6. 30 (K— “2:1, (3.: ‘%§7,) So '“xe 8%“qu sakﬁ-‘M
0% ‘HQ \Avwswoos eﬂu «\wm‘ x' 5
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3%) "' C‘ e “' 1 . Sway? N2 053m C1:"1‘Q\$CQ. mm _\.\A€
$o\o¥\m 09V OuC \NkA“ \Jo~\UQ ‘3“)me { 5 5 at w» = SW; ~16!” + C33 B3 \n3YEC\\vvx D, is om QESQMcAue 03? 4%: mﬁﬁx
(mi cm) W s is case a.“ “GM“ (mummy
owl $0 9. we}: ‘02 AM \uéc exﬁwduz (“PM 1).
We. weak He eéxmsymus o‘ SW D E “SM R E Wm 3. Find a general solution for the ODE
4y(5) + 831(4) + 53/" + 4y" + 83/ + 53/ = 0. (Hint: the function y(t) = 6“ sin(%) is one solution.) Wm M Ckuwéﬁtﬁév‘xc ea‘uckm is
*6“: $vﬁ+5v3 4 43+ (Zn Jr 5 3 O TMQ \Am)‘ ﬁfe/“S \)$ ‘A‘MAJY r: “\tib
OWQ No SOUH‘DAS- \A‘uwz (C— Phi \“k —-(-\ 2.‘\\ 2 CL +1v + 5 SkoAA he °~ guests? DQ “\\~Q 4. Find the solution of the heat ﬂow problem 3U 82’“- 8—t($’ 0:87;: (0<a:<7r,t>0)
M015): u,=(7rt) 0 (t>0)
“($70)=€_x (0<x<7r) “Q SUMO“ So\0'\'\o~f\ ogg JANE £ch AMa “ABS iS" -n‘rl', _ (mm): 2: one who , OT ‘0 36: {Cd 80 Arm ’M‘L OX 6 , wc amok
CW: %S:E Q—XS‘mLHQ A1. =—-,—%[ a \ *— {zwq o 1
K91: I ”x .. l - “X 3 «M - 8N9, WNW] — E. R» D + “e to (. \Xw I o a " 'L 1‘ '. ‘n V\K\AK,B .. 3‘an «“Q“L'\\n " “ go e’ S L “ 7K - “ m ”W" ~ “mom - Q [C 9- ( \ _ Zn _ ”‘1‘ _\ n TMOS Cw mix“ 6 ”7 wt 5. Find the formal solution of the problem (92H 8%; 5;($ay)+53§($7y)=0 (0<:v<7r,0<y<1)
U(0,y) =U(7r,y) = 0 (0 < y <1) %3($70)=03 ”($21):f($) (0<£D<7T) We use swwﬁ‘m .Q miésvs = Mx® =Xm\{L3\)
ﬁg ‘m’w ?DE‘ X” Y+ XY” = O ) Adds :9er
£31."; \< M xwvxm =0, \(I(0\=O.
X 3
Tye 0N; $0: X is {[email protected]“ \<X :0 mm, o u ‘ XW: CS‘VQ (0X\
wkiéx \/\0»$ 3% \ S\)W"\ X‘ a (V O ‘S;(X\ 2 E C“ (03‘ k“) 8““k“x\ ) C“: .33...“
‘ nu ‘ ncush n O 6. Let A be a square matrix, and let p be the characteristic polynomial of A. (a) If q is a polynomial such that q(A) = 0, show that q(/\) = 0 for. each eigenvalue A of A.
(b) Show that if A is diagonalizable then p(A) = 0. (M Svmsose «limit A is on eiaemwxoc 0Q Q . ...

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