MA 266 FINAL EXAM INSTRUCTIONS May 2, 2005

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1 MA 66 FINAL EXAM INSTRUCTIONS May, 5 NAME INSTRUCTOR. You mut ue a # pencil on the mark ene heet anwer heet.. If the cover of your quetion booklet i GREEN, write in the TEST/QUIZ NUMBER boxe and blacken in the appropriate pace below. If the cover i ORANGE, write in the TEST/QUIZ NUMBER boxe and darken the pace below. 3. On the mark-ene heet, fill in the intructor name and the coure number. 4. Fill in your NAME and OLD 9-DIGIT PURDUE ID NUMBER, not your new PUID number and blacken in the appropriate pace. 5. Fill in the SECTION NUMBER boxe with the diviion and ection number of your cla. For example, for diviion, ection 3, fill in 3 and blacken the correponding circle, including the circle for the zero. If you do not know your diviion and ection number, ak your intructor. 6. Sign the mark ene heet. 7. Fill in your name and your intructor name on the quetion heet above. 8. There are 5 quetion, each worth 8 point. Blacken in your choice of the correct anwer in the pace provided for quetion 5. Do all your work on the quetion heet. Turn in both the mark ene heet and the quetion heet when you are finihed. 9. Show your work on the quetion heet. Although no partial credit will be given, any dipute about grade or grading will be ettled by examining your written work on the quetion heet.. NO CALCULATORS, BOOKS, OR PAPERS ARE ALLOWED. Ue the back of the tet page for crap paper.. A table of Laplace Tranform can be found on the lat page of the quetion heet.

2 . If yt = in t i a olution of y + 9y = ft, then ft = A. in t B. A co 3t C. D. 5 in t E. 3 in t. If y = yx i the olution to then y = A. 4 B. 6 C. D. E. dy dx = 4xy, y = 4, + x 3. The general olution to x y + xy = e 5x i A. y = 5 e5x + c B. y = 5x e 5x + cx C. y = 5x e 5x D. y = ce 5x E. y = 5 x e 5x + c

3 4. The olution to the problem i xy + x 3 dx + x + y 4 dy =, y = A. x + 5 y5 = 5 B. x y + 5y 5 = 5 C. x y + x4 4 + y5 5 = 5 D. x y + 4x 4 + 5y 5 = 5 E. x y + x 4 + y5 5 = 5 5. The olution in implicit form of i: dy dx = x + 3y xy A. x + y = x 3 + C B. x + y = Cx 3 C. x + x 3 = y + C D. Cx = x 3 + y E. x + y 3 + xy = C 6. Which of the following bet decribe the tability of equilibrium olution for the autonomou differential equation y = y4 y? A. y = untable; y = and y = both table B. y = untable; y = table C. y = and y = both table D. y = table; y = untable; y = table E. y = table; y = and y = both untable 3

4 7. Solve the initial value problem y y = e t with y = a. For what value of a i the olution bounded i.e., not tending to infinity a t + on the interval t >? A. B. C. 3 D. all value E. no value 8. Initially a tank hold 5 gallon of pure water. A alt olution containing lb of alt per 3 gallon run into the tank at the rate of 5 gallon per minute. The well mixed olution run out of the tank at a rate of gallon per minute. Let xt be the amount of alt in the tank at time t. Find a differential equation atified by xt. DO NOT SOLVE THE EQUATION A. dx dt = 5 3 B. dx dt = 5 C. dx dt = 5 3 D. dx dt = 5 E. dx dt = 5 3 x 5+3t 3x 5+3t 3x 5+t x 5+3t x 5+t 9. The function y = t i a olution of the differential equation t d y dt tdy + y =. dt Chooe a function y from the lit below o that the pair y, y form a fundamental et of olution to the differential equation. A. y = t in t B. y = t e t C. y = t in t D. y = t E. y = t 4

5 . The larget open interval on which the olution to the initial value problem co t y + y = t y = ln 4 t t 3 i guaranteed by the Exitence and Uniquene Theorem to exit i A. π < t < π B. < t < π C. π < t < 3 D. < t < 4 E. 4 < t <. If yx i the olution of y y y = atifying y = and y =, then y = A. e B. e C. e e D. e E. e + e. The general olution yt of the differential equation y 3y + y = e t i A. yt = c e t + c e t + e t B. yt = c e t + c e t + e t C. yt = e t + e t + cte t D. yt = c e t + c e t te t E. yt = c e t + c e t + te t 5

6 3. The value of the contant r uch that y = x r olve x y + xy y = for x > are A. ± B. ±i C. ± D., E., 4. The proper form of the particular olution of the differential equation y + 3y + 3y + y = e t ued in the Method of Undetermined Coefficient i A. Ae t B. A co t + B in t C. At co t + Bt in t D. At e t E. At 3 e t 5. The invere Laplace tranform of A. 4e t + te t + e 3t B. e t + 3te t e 3t C. e t + 3te t + e 3t D. e t + 3te t + te 3t E. e t + 3te t e 3t i 6

7 6. The Laplace tranform of i A. F = + 4 B. F = C. F = + 4 D. F = E. F = 4 ft = t t τe t τ co τ dτ 7. Find the Laplace tranform of A. + e 4 3 B. + e C. + e D. + e 3 E. + e 3 ft = {, t <, t, t. 8. A ma weighing 6 lb tretche a pring ft. The ma i pulled down ft from the equilibrium poition, and then et in motion with a downward velocity of 8 ft/ec. Auming that there i no air reitance and that the downward direction i the poition direction. The gravity contant g = 3 ft/ec. Then, the amplitude of the ocillation i: A. B. C. D. E. π 4 7

8 9. The olution of the differential equation i A. u te t int B. C. e + u t t t+ D. u te t in t E. u te t+ int + y y + y = δt, y =, y =. The general olution to i y 4 + y + y = A. c co t + c in t B. c e t + c e t C. c co t + c in t + c 3 t co t + c 4 t in t D. c e t in t + c e t co t E. c e t + c e t + c 3 in t + c 4 co t 8

9 . The phae portrait for a linear ytem of the form x = A x, where A i a matrix i a follow y x If r and r denote the eigenvalue of A, then what can you conclude about r and r by examining the phae portrait? A. r and r are ditinct and poitive B. r and r are ditinct and negative C. r and r have oppoite ign D. r and r are complex and have poitive real part E. r and r are complex and have negative real part. The function x t determined by the initial value problem x = x x = x with initial condition x = and x = i given by A. x t = in t + co t B. x t = in t + co t C. x t = et + e t D. x t = co t E. x t = ie it ie it 9

10 3. Find the general olution of the firt order ytem 6 4 x = A x where A = given that ξ = i an eigenvector aociated to the repeated eigenvalue r = 4 for the matrix A, and that η = atifie A 4I η = ξ. A. c B. c C. c D. c E. c e 4t + c t e 4t t + e 4t + c t e 4t e 4t + c t + e 4t + c e 4t + c t e 4t e 4t e 4t

11 4. Find the olution of the initial value problem x = 4 A. B. C. D. E. e 3t + e 3t + e t + e 3t + e 3t e t e t e 3t e t e t x with x = 5. Conider the ytem x = α x α For what value of α i the equilibrium olution x = an aymptotically table node? A. no value of α B. α < C. α > D. < α < E. all real α

12 ft = L {F } F = L{ft}.. e at a 3. t n n! n+ 4. t p p > 5. in at 6. co at 7. inh at 8. coh at 9. e at in bt. e at co bt Γp + p+ a + a + a a a a b a + b a a + b. t n e at n! a n+. u c t e c 3. u c tft c e c F 4. e ct ft F c 5. fct c F, c > c 6. t ft τ gτ dτ F G 7. δt c e c 8. f n t n F n f f n f n 9. t n ft F n