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practiceExam1Sample2Solutions

# practiceExam1Sample2Solutions - Problem I[8 points If...

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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 Sammy-x 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): é-éfgv‘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 / (0-2) ’/ N t - on - .___ \ 5530325.. 4:60) + ( 23:? (o) + l J: (o 2 -.-_ 1 4. 0 “22:60.23 =- (3-014 “L2 Ema/F rum! (ID-0159787” ~ Em» 3 0.0007297--. 4' l (c) Repeat part(b) using a Taylor series expanded around x = 0.4. :HO-L) 1‘5 -¥Co.%)wCD~L3f%O-\t> + C?Lfi((o.%\ attest) = o- smut {as} = ~o~63r11F +1 §i’[9.q§: ——l-lS‘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' 7-2.6”): 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 60-0[_L£7€n)—33(0)4~1><1(0)3[03] 7 -; l+Co-t§[*4“’+lj:o4 +2. A 3 git—é Xfo'l): xwutﬁ- [ort3£x(o.ﬁ_uyto.0 -+X(a-t')j = ought CULQE o.§—z.g+O-S‘tz7:/O~€3‘(Z l: )4an l 360.2) z: 3604» (mot—435"“)“8?{”“)”"l0“53“433 '4’ 3 ' : (9.7+ an, aﬂwzﬁwolmz-t 0-3010 ﬁlo-473': 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/L-U‘Wb 102" 9 ...
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