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Unformatted text preview: Problem I: [8 points] If possible, rewrite the systems of simultaneous equations deﬁned by in a matrix—vector form that could be solved using Gauss elimination. If this task is not possible, state
why. x+3y=2xy+7ym2
(a) 3zm2x—y
x+lZz=2y+12 Nof $055M; in km. .+Lf xl +32:3 =x4 +2
x3 —2x2 =17
x3+2x1“12x4 =0
x3 +2362“):4 =20 (b) ['0 3‘4 x: 2' o «z I 0 XL _ r7
1 O i VIZ X3 O
O 1 i *i M 9'0 Problem 2: [8 points} For each of the ODEs deﬁned below, calculate y(1) by ﬁnding the ODE’s
exact solution. (a) g = x2 + we” with y(0) = o . wow ('.><’+w"‘>+c = =3: a W 509:0 re czT . wl
2:) an) : é—wm‘m +971“: 2.3m (b) 3y”3y’—18y=0 with y(0)=0 and y'(0)==1.
Haj Sammyx gmw m2. ___ M __ :0 =9 Cm._3>[_'m+2_> ~.—. 0 I 1x m and : Ad + ’32:” '4— 2
T05 (web—:6: A+B .e Ez—A
g’wb: I: 3A 213
“:9 I =3A+QA=€A =3 Aag': )”E=~;. 4»;
$0): ééfgv‘g; a“; = ’3 new + ‘ Rwy" Problem 3: [7 points] This problem considers the Taylor series expansion of the function f(x) = er: . (a) Differentiate f(x) to find f '(x) and f ”(x). L
_)(
5100 2 e
L
—)<
$100 = — 31X 9/ X 2
2. 1 ._r
x .«Zx
owtx) = v29, + 428% = (thz 1% "3’ I
(b) Use the results from (a) to approximate f (0.2) using a second order Taylor expansion
expanded around 3: = 0. State the error between your calculated result and the exact value
of f (0.2) to 3 signiﬁcant ﬁgures. 1
/ (02) ’/
N
t  on  .___ \
5530325.. 4:60) + ( 23:? (o) + l J: (o
2
._ 1 4. 0 “22:60.23 = (3014 “L2 Ema/F rum! (ID0159787” ~
Em» 3 0.0007297. 4' l (c) Repeat part(b) using a Taylor series expanded around x = 0.4. :HOL) 1‘5 ¥Co.%)wCD~L3f%O\t> + C?Lfi((o.%\ attest) = o smut {as} = ~o~63r11F +1
§i’[9.q§: ——llS‘2>°1'
a 5(a).) 1y @«MS‘KOXZ
~3— ( 4 Emma (9— OOQQ , Problem 4: [8 points] Your roommate claims that the following ODE describes the stock
market, 4 2
145+ 37r3zi§+ Z3 sin(t)=121,‘.
dt dz: (a) Is this ODE linear or nonlinear? If it is nonlinear, state which terms make it nonlinear. i‘ximLCMLA/‘Jb'fC/Oi; ZZ/IM 23 (b) If this ODE is converted to a system of ﬁrst order ODEs, how many equations will the system have? .
1+ + I (c) Convert the ODE to a system of ﬁrst order ODEs. 21:2 21’: z; I 2222’ 22: :23, + 23:2.” “’5 23 =2? 3 3 2992!” Z; :  37;“; 2’23 —— zlmhft) + [215' (d) The initial conditions your roommate gives for the equation are 2(a) = l, 2'05) 2 2 ,
z”(7r) = 3 , z'”(:r) = 4. Use these initial conditions to deﬁne initial conditions for the
system of ﬁrst order equations you deﬁned in (c). 21513“): 1'
72.6”): 2' Problem 5: [14 points] (a) Give an example of a 4X4 matrix that is upper triangular. +2. (b) Give the numerical value of matrix element an,” if the matrix A is the 20X20 identity
matrix. ‘ 0
+2 (0) If B is a 6X7 matrix and BC = D is well deﬁned, how many rows do the matrices C and D
have? '8 c a ‘D
(Mum (gm Cleaﬂ'7rows
'Eha 6% (d) If E is a square matrix, what is the simplest way to rewrite EIE'IEIZE'II'I? e—  l
ELEEIEI :EE'EE : I it (e) If F is a square matrix with eigenvalue 9L, is 27L an eigenvalue of the matrix G : 2F? State
your reasoning. (hit? a (mum)? «a axles?) Yip.
+2, (f) 11°F is a square matrix with eigenvector 17 , is 217 an eigenvector of the matrix G = 2F?
State your reasoning. am uWroﬂ— 0.32F i  Problem 6: [15 points] This problem examines the system of ODES
ix— : x — 4y + x3 g dt 01? 3 2 — r— —4 — + 2x
F d: y y y
g with initial conditions x(0) =1 and y(0) =1 .
l (a) Calculate the solution of the ODEs at t a: 0.2 using two steps of Euler’s method with a time step of O. 1. Show your work.
git XCOnl) s. xto)+ (mum .45 £03 +73603]
: 1+ (0~t)[§.~.qj 2.. (9»?
(54“) 15(0) .L 600[_L£7€n)—33(0)4~1><1(0)3[03]
7 ; l+Cot§[*4“’+lj:o4 +2. A 3
git—é Xfo'l): xwutﬁ [ort3£x(o.ﬁ_uyto.0 +X(at')j
= ought CULQE o.§—z.g+OS‘tz7:/O~€3‘(Z l: )4an l 360.2) z: 3604» (mot—435"“)“8?{”“)”"l0“53“433 '4’ 3
' : (9.7+ an, aﬂwzﬁwolmzt 03010 ﬁlo473': last”). (b) Calculate the solution of the ODEs at t : 0.2 by using a single step (of length 0.2) of.
Heun’s method. Show your work. $91, an». 1’ XEWWW
K v 3‘:g(x.gj.,{~) .+3 rpwfdﬁr SLY) X" _; Xfo) + (0.2,); (1,!) o) :2; l 4 (0L2:)(—2/) 2&5
I 39: 369+ [oiDg C1350) 2 1+ (010(4) = OWL. g Cﬂﬁf‘wﬁr‘ 8LT; (’~ ' ,
X0952) : X60 + 3;)thaslﬂ)+7950~5JMJ0¢239 +1 (c) If you recaloulatod the solution at I: 0.2 using Heun’s method with a time step of 0.02,
' how much more accurate would the solution be than your solution from (b) that used a _+, 3
time step of 0.2? HWWWUWQ —} vmria Ml) _‘ i’w 1» L310 e.) Wag/ULm/LU‘Wb 102" 9 ...
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 Spring '07
 Gallivan

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