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2nd midterm solutions

# 2nd midterm solutions - l Consider the system xi = 1 — my...

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Unformatted text preview: l. Consider the system: xi = 1 — my y' := I — y3. Determine its singular (equilibrium) points and classify each, insofar as possible, using linearizatiOn. In particular, classify each equilibrium point as stable or not stable. For each equilibrium point, deten‘nine in addition if it is a focus. node, or saddle, if you have the information to do so. If one cannot detennine the type from the linearization, say so, and indicate, if possible, the alternatives. No sketch is required. The Singulm (to!) and (—ldl] is, \$3 ’3 '7‘ a: s: ' ‘33” . AL (1):] 4‘0») \$3611.) "I “l 3x01“) 530.10 I '3 Hm: p: 'lr3=*‘f (5:613031‘0390 =7 . The; poghujf-l‘s ah ‘l'kn Wllb P1341). Hana}, anoo (bl) is shit-8:. -— became. p<o~ [out wi‘l'kau'l My. mung-‘35:“: at dad: lemon.) .‘l,’ CI”) is a \$4109.: £30.44 or a Club-9‘2 {W or- 6 (""0 {x0543 EJI'IJ'I) 1" 1 3.9V!) ant-m) ’ 1 -—:s Hm pz-z , ff"! . Smut g<OJ (firm is a. aeolian, Soda-04! poida w unduler . 2. Consider the linear system :c’ _ —4 4 it y“ w 1 "4 y ’ If A is the matrix of coefﬁcients. det(A — M) = (A + 2)(A + 6). Classify the singular point at the origin. (Is it stable or unstable, a focus, node, saddle, etc?) Sketch the phase portrait near the origin. The sketch can be rough but it should have the right qualitative features, and. if there are lines through eigenvectors in the picture, they should point in roughly the correct directions. You do NOT need to compute and sketch z— and y— isoclines. The, eigenvalue am. Fly-“'2 and. \$1=-6. The, origin. L5 cm (asam'o‘lbhnggj) shtle 5»th poii. ‘1. Th.» engeémr- a53ocia'lpd lo 95.1-1 {5 V‘: I Tim. eigemceel-or quOCitIDJ 4'0 AL: —6 'rs [—71 All Ryder—res Q‘Ppusadn Cop) langmi 4° Vt “apt. ‘hﬂQ “agreeing dang UL 5 o A A. A I- ? A. .5 b 83, = (‘ j - (V33) eI — «3,8279, : I - e... __ e2 . Let (egg) : my] + 3323,12 + 373313 + \$41“ deﬁne an inner product on R3 with norm “1'” = (33533). Note: This is NOT the dot product! Let “2 -— 11-3 = EDI—‘00 0 1 1 U p—lp—lg—Ap—A The Gram-Schmidt method applied to the ﬁrst two vectors in {m , ’02, 93} in that order give 1 0 1 U 1 A 1 62: 0 a 82:“ U 1 1 I \/5 0 1 (a) Find the last vector ég by the Gram-Schmidt method. (b) Supppose w is a vector satisfying (w,é1) = 3, <10,é2) = 2V3 and ||w|| = 6. Find the vector T which is the best approximation to w in Span{é1,é2} in the some of minimizing the magnitude of the error llw — T||2. What is the error [Iw -— T||2? l 0 r5 0 0 ° } i '3 .. z =M-(‘3i'3’r °i 6' a o 0 l _3 "3 z 1. E3 = 93- -.-. J" 2 Smith “('39 d'5)ii=q5*3[ﬂl*3 u (\ LE- pa- J 9) 4.. h. _E. «o; (J \f If)» n on r—H 0—50 Nu._..-/ .5. M slaw _ a - '- II P” ” "‘ w 4. Let f(\$) = 1 w 3; on (0,1). Consider the following series 51m) : %+ : Til—Tr sin(2mrm) Fourier“ Sen'e 3 52m) = %+ : é cos(3n\$) N one, 53m) z 52; 5111mm) H 135(3) 54‘“ = 1”) erect?) warm] mam if: +ﬁ1/2 +U:—:L1‘i7LTI)’-’] Si“ vigil”) ' 0 Fish?) 2913 31, || NIH + M8 .5,“ E II 0M8 MIN, I. 1r (a) One of these series is the ordinary Fourier series F S ( f ] of f for the extension of f with period 1, one is the half-range cosine series HRCU'), One is the quarter-range cosine series QRCU), one is the half—range sine series HRSG}, one is quarter-range sine series QRSU‘), and one is none of these. By looking at periodicity and series form, determine which is which. (b) Graph the extensions of f corresponding to the half range sine series (HRS) and the quarter range sine series (QRS) series. 9 H25 exicmSion 4.: [-ml ﬁesta has pared. 2, I l l 1 Exiension 'izo [-1.1] 6'“) a carceSponalt-lj lro east?) The. Q2514!) had I fainted 4. 5. Show that phase trajectories of the system :r.’ : —8Iy, y’ 2 I2 lie in curves of the form 83,12 + :1".2 : c. Sketch a phase portrait with severa‘i trajectories. Indicate all singular (equilibrium) points of the system. Indicate the direction of motion of the trajectories. Are trajectories of this system periodic or not? 5’; doc. I} 1. 2+ ‘ﬁa (=5) Sgd3=-xciac. 6:) ‘15=-’/,_x L {=9 831 +21?- :6. Every Poini' on {ht g-axm (Every poin'L wi'i'h 22:03 is at sinauiM. point. 5%— : x2 70 »?ur' .JLQ xgo J do 'ilraaedbries aloud”: man-P (.2pr a: Tmydories am not pmtodie. A iraaocfbrd cam new mass "the g-axis (ram aﬂﬁ Pot-J5 on {:he g—axis cw: ctnMa-A poivd'a_ I Phase. i-rogedories lie in Jrkn. eUipJeS guqu—z: C. & yvouct‘s : Smaoiwt paid-4 6. Find the half range sine series of cosh) on [0, 1r). 9° 1 HRS = :bnsinhx) J 5., = % L magnum; .n, 1 % fmfaﬂdx :0 4 n=1 D b" = J5. [Sin (n +\\x. +55": (ii-ﬁx] AL 1-. 1 o -.L “(max * coda-[)1] n71 1‘ nu n-I 3 O O , “=1 = “‘4 in. ,Lq 1 2n 'n [I ""' 3 r‘ 31 O J (17.1 — { 1n [\ + 00"] 'ﬂtn"-0 m 05 2nLHL-0n] - _ Z 8k sinuhx. : -...——-—-—v—-- ‘5 C " Thus HRS E “(ﬁt-L) m ax) Rd ———"'*—Tr CLIP”) D ...
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