MA FINAL EXAM INSTRUCTIONS

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1 MA 33 FINAL EXAM INSTRUCTIONS NAME INSTRUCTOR. Intructor nme: Chen, Dong, Howrd, or Lundberg 2. Coure number: MA SECTION NUMBERS: 6 for MWF :3AM-:2AM REC 33 cl by Erik Lundberg 7 for MWF :3AM-:2AM MATH 75 cl by Min Chen 72 for MWF :3AM-2:2AM REC 33 cl by Erik Lundberg 73 for TR :3PM-2:45PM UNIV 7 cl by Suchun Dong 74 for TR 2:AM-:5PM UNIV 7 cl by Suchun Dong 75 for MWF :3PM-2:2PM UNIV 29 cl by H Howrd 76 for MWF 2:3AM-:2PM UNIV 29 cl by H Howrd 4. TEST/QUIZ NUMBER i: if thi heet i yellow 2 if thi heet i blue 3 if thi heet i white 5. Sign the cntron heet. 6. Lplce trnform tble nd formul heet on Fourier erie nd ome PDE re provided t the end of thi booklet. 7. There re 2 quetion, ech worth 5 point. Do ll your work on the quetion heet. Turn in both the cntron form nd the quetion heet when you re done. 8. Show your work on the quetion heet. Although no prtil credit will be given, ny dipute bout grde or grding will be ettled by exmining your written work on the quetion heet. 9. NO CALCULATORS, BOOKS, OR PAPERS ARE ALLOWED. Ue the bck of the tet pge for crp pper.

2 . Conider the Euler eqution x 2 y + αxy + βy = Find ll mong the following condition on α nd β o tht ll olution re bounded x? A. α <, β. B. α =, β >. C. α =, β D. A nd C E. A nd B 2. Find the vlue of n=,n=odd n 2 = Hint: Evlute the following Fourier erie t x = uing the fct tht the Fourier erie of period 4 for the function f(x) = 2 x defined on ( 2, 2) i + 8 π 2 n=,n=odd co( nπx 2 ) n 2. A. π 3 B. + 8 π 2 C. D. E. π 2 8 π π 2 3. Which of the following (if ny) i not n eigenvlue-eigenfunction pir for the eigenvlue problem y + λy =, y () =, y (3π) =? A. {λ = 4 9, y(x) = co( 2x 3 )} B. {λ = 4, y(x) = co( x 2 )} C. {λ = 4, y(x) = co(2x)} D. {λ = 9, y(x) = co(3x)} E. Thee re ll vlid eigenvlue-eigenfunction pir. Pge 2

3 4. Which i the olution for A. 2e 3πy inh(3πx) B. 2e 3πy in(3πx) C. 2 inh(3πy) in(3πx) D. 2 inh(3πy) co(3πx) E. none of the bove. 2 u x + 2 u =, 2 y2 x, y u(, y) =, u(, y) = u(x, ) = 2 in(3πx), lim u(x, y) =? y 5. If y + 3y + 2y = δ(t 2) with y() =, y () =, then y(4) i A. e 2 e 4 B. e 2 2e 4 C. e 4 e 6 D. e 2 E. 2e 4 e 2 6. If y(t) olve the initil vlue problem { t/2, t < 6 y + y =, 3, 6 t y() =, y () = for t, then { A. y = in(t) + t t < 6 in(t) + 3 t 6 2 { B. y = in(t) + t t < 6 in(t) + 3 t 6 { C. y = in(t) + t t < 6 in(t) + in(t 6) + 3 t 6 { D. y = co(t) + t t < 6 co(t) + in(t 6) + 3 t 6 { E. y = in(t) + t t < 6 in(t) + in(t 6) + 3 t 6 Pge 3

4 7. If L{f(t)} = , ( 2 + 4) then wht i the vlue of f( π 2 )? A. 6 B. 3 C. D. -2 E The indicil eqution for 2x 2 y xy + ( + x)y = t x = i A. 2r 2 3r + = B. 2r 2 3r = C. r 2 3r + = D. r 2 r + = E. 2r 2 r + = 9. Conider the ytem x = Ax, nd uppoe the eigenvlue re r = 2 ± 3i. Then clify the origin in term of the phe portrit. A. pirl B. center C. proper node D. ddle E. improper node Pge 4

5 . Find the Lplce trnform of when t < π, f(t) = t π when π t < 2π, when t 2π. A. e π 2 e 2π 2 πe 2π B. e π 2 e 2π 2 C. e π 2 e2π 2 πe2π 2 D. ( e π e 2π) E. 2 e π 2 3 e 2π 2 πe 2π. Suppoe X(t) i fundmentl mtrix for the homogeneou eqution x = Ax, nd uppoe tht x = X(t)c(t) i olution of the nonhomogeneou eqution x = Ax + g(t). Which of the following i correct eqution for c? A. c = Xg B. c = X g C. c = Xg D. c = X g + Xg E. c = X g 2. Conider the ytem: x = [ c 2 ] x. For wht vlue of c i the origin n ymptoticlly tble pirl? A. c < B. 5/2 < c < 2 C. c < D. < c < 3/2 E. c < 9/4 Pge 5

6 3. (Hint: By oberving the correct form of the olution, you hould be ble to find the nwer for thi [ problem) ] 2 The mtrix h eigenvlue nd, nd their correponding eigenvector 3 [ 2 ] [ ] re repectively nd. Which of the following i the olution X to the initil 3 vlue problem X = [ ] e A. t + e t + te t 2e t 2e t + te t [ ] 3e B. t + e t te t e t e t + te t [ ] e C. t + e t 2te t 2e t + 2e t 2te t [ ] e D. t e t te t 3e t 3e t te t [ ] e E. t e t 2te t 3e t + 3e t 2te t [ ] [ e t X + 5e t 4. The generl olution to the ytem x = ( 2 3 ( ) ( ) ( ) 2t A. x = c e 2t + c 2 e 3t 2 + 2e t ( ) ( ) ( ) 2t B. x = c e 2t + c 2 e 3t + 4e t ( ) ( ) ( ) 2t C. x = c e 2t + c 2 e 3t + e t ( ) ( ) ( ) 2t D. x = c e 2t + c 2 e 3t + te t ( ) ( ) ( ) 2t + E. x = c e 2t + c 2 e 3t + 4e t ] [, X() = ) ( 4t x + 2e t ) i ] Pge 6

7 5. For which vlue of λ doe the following eqution h nontrivil olution u xx + u yy + λu = u(, y) = = u(, y) u(x, ) = = u(x, b) Find the correponding olution. Hint: Ue eprtion of vrible. x, y b < y < b < x < A. The poible vlue of λ re : m2 π 2, m =, 2,... with correponding olution 2 in( mπx B. The poible vlue of λ re : n2 π 2 in( mπx b ) in(mπy ); m =, 2,... b 2, n =, 2,... with correponding olution ) in( mπy ) n =, 2,... b C. The poible vlue of λ re: m2 π 2 + n2 π 2 ; n =, 2,..., m =, 2,... with correponding 2 b 2 olution u mn = in( mπx D. There re no uch vlue of λ. ) in(mπy ); n =, 2,..., m =, 2,... Pge 7

8 6. For the eqution x 2 y xy + (x 2 )y =, the correct form for 2 linerly independent 6 olution with x > re A. y = x 2 ( + n= nx n ),y 2 = x 3 ( + n= b nx n ) B. y = x ( + n= nx n ), y 2 = y (x) ln x + x ( + n= b nx n ) C. y = x /2 ( + n= nx n ), y 2 = x /2 ( + n= b nx n ) D. y = x( + n= nx n ), y 2 = y (x) ln x + x( + n= b nx n ) E. y = x co( 3 ln x/2)( + n= nx n ), y 2 = x in( 3 ln x/2)( + n= b nx n ) 7. Given then wht i the vlue of L{f(t)}? t < f(t) = t t < 2 t 2 A. e 2 B. e C. e 2 D. e 2 e 2 2 e 2 + e e e 2 2 e 2 2 E. + e e 2 2 2e 2 Pge 8

9 8. Then y ( π 2 ) =. y + 2y + 2y = δ(t π 6 ) in t y() = y () = A. e π 2 B. 3 2 e π 2 C. 2 e π 3 D. e π e π 3 E. e π e π 3 9. Let u(x, t) tify the het eqution Then u( 2, ) = A. B. 2 C. + e π2 D. 2 e π2 E. + 2e π2 u xx = u t, < x <, t > u(, t) =, u(, t) = 2 u(x, ) = 2x + in(πx). 2. The pproximte vlue of the olution t t = 2 of y = t + 6y, y() = 2 i evluted uing the Bckwrd Euler method with h =. Recll: The formul of Bckwrd Euler method i t n+ = t n + h; y n+ = y n + hf(t n+, y n+ ) for n =,, 2,. The pproximte vlue of y(2) i A. 2 B. 3 C. 5 D. 8 E. 3 Pge 9

10 Formul heet Fourier erie: For 2L-periodic function f(x), the Fourier erie for f i where for n =, 2,, 2 + n= n co nπx L + b n in nπx L, = L L f(x)dx, n = L L f(x) co nπx L dx, b n = L L f(x) in nπx L dx. Het eqution : The olution of the het eqution α 2 u xx = u t, < x < L, t >, tifying the (fixed temperture) homogeneou boundry condition u(, t) = u(l, t) = for t > with initil temperture u(x, ) = f(x) h the generl form u(x, t) = n= c n e n2 π 2 α 2 t/l 2 in nπx L, where c n = 2 L f(x) in nπx L dx. Het eqution 2: The olution of the het eqution α 2 u xx = u t, < x < L, t >, tifying the inulted boundry condition u x (, t) = u x (L, t) = for t > with initil temperture u(x, ) = f(x) h the generl form u(x, t) = c 2 + c n e n2 π 2 α 2 t/l 2 co nπx L, where c n = 2 L n= f(x) co nπx L dx. Wve eqution: The olution of the wve eqution α 2 u xx = u tt, < x < L, t >, tifying the homogeneou boundry condition u(, t) = u(l, t) = for t > nd initil condition u(x, ) = f(x) nd u t (x, ) = g(x) for x L h the generl form where c n = 2 L u(x, t) = n= in nπx L ( c n co nπαt L + k n in nπαt ) L f(x) in nπx L dx nd k n = 2 nπα g(x) in nπx L dx. Lplce eqution: The olution of the Lplce eqution u xx + u yy =, < x <, y b, tifying the boundry condition u(x, ) = u(x, b) = for < x < nd u(, y) = nd u(, y) = f(y) for y b h the generl form u(x, y) = n= c n inh nπx b in nπy b where c n = 2 b inh( nπ b ) b f(y) in nπy b dy. Pge

11 Figure : Lplce Trnform Tble f(t) =L {F ()} F () =L{f(t)}. 2. e t 3. t n n! n+ 4. t p (p> ) 5. in t 6. co t 7. inh t 8. coh t 9. e t in bt. e t co bt Γ(p +) p b ( ) 2 + b 2 ( ) 2 + b 2. t n e t n! ( ) n+ e c 2. u c (t) 3. u c (t)f(t c) e c F () 4. e ct f(t) F ( c) 5. f(ct) 6. t f(t τ) g(τ) dτ ( ) c F c F () G() 7. δ(t c) e c c> 8. f (n) (t) n F () n f() f (n 2) () f (n ) () 9. ( t) n f(t) F (n) () Pge