s04_DE_final - INTRODUCTION TO DIFFERENTIAL EQUATIONS FINAL...

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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 vole-me 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 O - v—— + c F+L PH. ELL,E --V f A(Qe"\:w'—€ -va 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‘H-x (2'- L ’ (Fiflt Law 707 EL: L (l—e, V) t_ FJrL. FH- __ \ Y’L VL "(FH’I‘ E/v 2 - —" " ' "' e» Fi’L ‘1»... .l : e, " '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 steady-state; 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 all-cases 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 A: x: — ("Tl-)1 Snacc, L:\ X{X\’ S\\'\ "Trx ’4 :\I 2I ‘ a w- mmW=o m Y: w (amino _ A“ 63:4773 + 6’ 3n7|3 J (4,. {ma}: (An 630173 * Q“ c'34n]> 5‘“ anx W 3v)“ '2“ ,9 U (X,‘J\ : Z (An er 3 ‘_ (3" e “3) S\V\ “fix 8M QWX ‘ “(x/0) 1‘- 0” \ “WK 7;) A‘L‘rfi‘b‘ \ g' (An’r Vb.) s h A”? an :0 , C“ a“ OWU“ -6nn‘ A4 : — fiq e For will Anz<bn=o 1 l l For ; v '65“ ‘ —~. ,1 " — V\ 02' ‘6 +\ :\ :1) Ybl- !_64 1T7- I‘e.'2fi1 _ vénfil At : e A .4 .l s -— Izrr1 1 ‘ 8’44'” amt e. -\ c '\ 6113 mm T261 4- e ‘ $\\n QWX um): " -\ “c 5 €,11“L 2 \2117- 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» ’z-r 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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