MATH 251Fall 2004Exam 1Oct 12th, 2 0 04NAME :ID :INSTRUCTOR :There are 12 questions on 10 pages. Please read each problem carefully before startingto solve it. For each multiple choice problem 4 answers are given, only one of which iscorrect. Mark only one choice. For partial credit questions, all work must be shown -credit will not be given for an answer unsupported by work.NO CALCULATORS ARE ALLOWED.PLEASE DO NOT WRITE IN THE BOX BELOW.1:2:3:4:5:6:7:8:9:10:11:12:Total:MATH 251 Fall 2004 Exam 11. (4 points) Identify each of the following equations as linear or non-linear and alsodetermine their order.(a) ydydt= t(b)d2ydt23+dydt3+ y3= t3(c) sin(t)dydt+ t5y = (1 − t2)d2ydt2(d) (1 + y)sin2t + (d3ydt3+ y)cos2t = 12. (6 points) The initial value problem(4 − t2)y′+ ln(t)y = sin(t) y(1.33) = 3.14159is certain to have a unique continuous solution on the interval(a) (−4, 4)(b) (−2, 2)(c) (0, 2)(d) (0, 3.14159)Page 2 of 10MATH 251 Fall 2004 Exam 13. (6 points) The general solution to the equationy′′+ 6y′+ 9y = 0has the form(a) ce−3t(b) c1e−3t+ c2e−3t(c) c1e−3t+ c2te−3t(d) c1e−3t+ c2e−3t24. (6 points) If y(t) is the solution to the initial value problemy′= y(y2− 4) y(0) = 3.Then limt→∞y(t) =?(a) 3(b) 2(c) −2(d) ∞Page 3 of 10MATH 251 Fall 2004 Exam 15. (6 points) Let y1and y2be two solutions to the linear equation2t2y′′− ty′− y = 0.Then the Wronskian of y1and y2must be a constant multiple of(a)√t(b) t(c) 1(d) t26. (6 points) Solutions to(x cos y)dydx= sin yare,(a) y(x) = tan(x + C)(b) y(x) = sin−1(−1x+ C)(c) y(x) = sin−1(Cx)(d) y(x) = sin−1(ln(x) −x2+ C)Page 4 of 10MATH 251 Fall 2004 Exam 17. (10 points) Solve the i nitial value problem.t3y′+ 4t2y = e−ty(1) = 0Page 5 of 10MATH 251 Fall 2004 Exam 18. (10 points) A tank has 100gal of water and 100lb of salt mixed in it. Water entersthe tank at the rate of 3gal/min with concentration of salt in it, at time t given bye−tlb/gal. A well mixed solution leaves the tank at the same rate of 3gal/min. Find aformula for the amount of salt in the tank at a ny time t.Find also the eventual concentration of salt i n the tank.Page 6 of 10MATH 251 Fall 2004 Exam 19. (12 points) Show that the equation(4xy − 3)y′+ 2(y2+ x) = 0is exact.Find the general solution.Page 7 of 10MATH 251 Fall 2004 Exam 110. (10 points) Solve the following initial value problems:(a)y′′+ 2y′+ 5y = 0, y(0) = 0, y′(0) = −1.(b)y′′+1ty′= 1 y(2) = −2ln(2), y′(2) = 2.Page 8 of 10MATH 251 Fall 2004 Exam 111. (12 points) Find α so that the solution to the initial value problemy′′+ 3y′− 4y = 0, y(0) = α, y′(0) = 1,converges to 0 as t → ∞.Page 9 of 10MATH 251 Fall 2004 Exam 112. (12 points) Given that y1= t is a solution to the equationt2y′′− 4ty′+ 4y = 0,use the method of reduction of order to find a second solution which is not a constantmultiple of y1.Show that the two solutions above are linearly indepen d ent on the interval (0, ∞).Page 10 of
View Full Document
Unlocking...