UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION, APRIL 13, 2012 Duration: 2 and 1/2 hours Second Year MIE
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1 UNIVERSITY OF TORONTO FACULTY OF APPLIED SCIENCE AND ENGINEERING FINAL EXAMINATION, APRIL 3, 202 Duration: 2 and /2 hours Second Year MIE MAT234HS - DIFFERENTIAL EQUATIONS Exam Type: B LAST NAME: as on your T-card YOUR FULL NAME: STUDENT NUMBER: SIGNATURE: Examiners: N. Hoell M. Pugh No Calculators Permitted. INSTRUCTIONS: Make sure your exam contains 24 pages. There are 20 points possible. Attempt all questions. Do not tear any pages from this exam. You can use the backs of pages for scratch work. You have been provided with a sheet of formulas that you may use during the exam. of 24
2 . 7 points Classify and solve the following initial value problem y + 2y = 2 t e 2t, y0 =. linear/nonlinear/homogeneous/nonhomogeneous/the order is: 2 of 24
3 2. The equation u tt + cu t + ku = a 2 u xx + F x, t where a 2 > 0, c 0, and k 0 are constants, is known as the telegraph equation. It arises in the study of an elastic string under tension. The term F x, t represents external forcing. a 3 points Assuming ux, t = XxT t is a solution of the unforced telegraph equation, find the ODEs that Xx and T t must satisfy. b 0 points Find the general solution of u tt + 2 u t + 2 u = 3 u xx for all x 0, π, t > 0 u0, t = uπ, t = 0 for all t > 0 This is the only problem on the exam where you are expected to do separation of variables to find a solution. If you find yourself starting to do separation of variables for some other problem, you re working too hard. c 4 points What is the solution that has initial displacement ux, 0 = 3 sinx and initial velocity u t x, 0 = 5 sinx? You can answer this question without having done part b if you make some good choices. Or you can just use your answer to part b. 3 of 24
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7 3. Consider the ODE. y y + lnx x y = 0 a 3 pts Verify that y = 4 and y 2 = lnx are both solutions of the ODE. b 3 pts Is y = y + y 2 a solution of the ODE? Why or why not? c pt Find a third solution of the ODE. 7 of 24
8 4. 2 points On the next page, you re given a collection of linear systems of ODEs. Below are four phase portraits. Above each phase portrait, write the letter of the corresponding linear system. This is system: This is system: sect_3p3_example_3.pdf sect_3p5_example_.pdf This is system: This is system: sect_3p4_exercise_3.pdf sect_3p3_example_5.pdf 8 of 24
9 Consider the system x = Ax where the matrices A are given below. system letter A A = 2 4 /2 B 2 5 A = 2 C A = 3/8 3/4 /4 /4 D A = 4 /2 2 E 2 0 A = 0 2 F 2 5 A = 2 G 0 A = 0 H A = /4 3/4 /4 3/8 4 λ = 3, v = 2 + i λ = i, v = 6 λ = 7/4, v = 4 λ = 0, v = λ = 2, v = 2 + i λ = i, v = λ =, v = 0 6 λ = /8, v = 2 λ 2 = 0, v 2 = 2 i λ 2 = i, v 2 = λ 2 = /8, v 2 = 2 2 λ 2 = 3, v 2 = λ 2 = 2, v 2 = 2 i λ 2 = i, v 2 = 0 λ 2 =, v 2 = λ 2 = 7/4, v 2 = 2 9 of 24
10 5. 5 points Consider the linear system of first-order differential equations d x dt = A x where A is a matrix with constant entries. Show that if v is an eigenvector of A with eigenvalue λ then x = e λ t v is a solution. 0 of 24
11 6. 7 points Consider the forced damped oscillator y + 2 y + 5 y = cost. The general solution y is the sum of two functions: a transient solution y c which decays to zero as t plus the forced response Y which is an oscillation at a certain amplitude and frequency. What is Y? What is its amplitude? of 24
12 7. 7 points Consider the heat conduction problem Define the average heat at time t by u t = α 2 u xx for all x 0, L, t > 0 u x 0, t = 0 for all t > 0 u x L, t = 0 for all t > 0 ux, 0 = u 0 x for all x [0, L] At = L L 0 ux, t dx. Use the heat equation and the boundary conditions to show that da/dt = 0. How is At related to the initial temperature distribution u 0? 2 of 24
13 8. 6 points When Zal was courting Rudaba, he wanted to show his love for her by tuning his oud so that it would speak with the same voice as hers: the first string on his oud would have the same natural frequencies as the first string on hers and so on. The strings on his oud were twice the length of the strings on her oud. They re made out of exactly the same material. How should the tension in the strings of Zal s oud be related to the tension in the strings of Rudaba s oud? 3 of 24
14 9. 2 points These are multiple choice problems with no partial credit. a Consider the heat conduction problem: u t = α 2 u xx for all x 0, L, t > 0 u0, t = 0 for all t > 0 ul, t = 0 for all t > 0 One of the following series is the general solution of this problem. Circle it. c n e αnπ/l2t sinnπx/l n= c n e αnπ/l2t sinnπx/l n= c n e αnπ/l2t sinnπx/l c n e αnπ/l2t sinn2πx/l n=2 n= b Consider the heat conduction problem: u t = α 2 u xx for all x 0, L, t > 0 u x 0, t = 0 for all t > 0 u x L, t = 0 for all t > 0 2 One of the following series is the general solution of this problem. Circle it. c n e αnπ/l2t cosnπx/l n= c n e αnπ/l2t cosnπx/l n=0 c n e αnπ/l2t cosnπx/l n= c n e αnπ/l2t cosn2πx/l n= 4 of 24
15 c Consider the heat conduction problem: u t = α 2 u xx for all x 0, L, t > 0 u x 0, t = 0 for all t > 0 ul, t = 0 for all t > 0 3 One of the following series is the general solution of this problem. Circle it. c n e n= c n e n=0 2n+ π α 2 2n+ π α 2 L 2t cos L 2t cos 2n + π x c n e 2 L n= 2n + π x c n e 2 L n=0 2n+ π α 2 2n+ π α 2 d Consider the wave propagation problem: u tt = a 2 u xx for all x 0, L, t > 0 u0, t = 0 for all t > 0 ul, t = 0 for all t > 0 L 2t sin L 2t cos 2n + π x 2 L 2n + π x 2 L One of the following series is the general solution of this problem. Circle it. n= n= c n cos anπ L t + k n sin anπ L t c n cos an2π L t + k n sin an2π L t n= n=2 c n cos anπ L t + k n sin anπ L t sinnπ x L sinn2π x L sinnπ x L c n cos anπ L t cosnπ x L + k n sin anπ L t sinnπ x L 4 5 of 24
16 0. Consider the initial value problem 0 0 d dt x = a 0 pt Solve the initial value problem. b pt As t, what does the solution do? x, x0 = 6 of 24
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19 . A shoe is thrown straight up into the air, reaches its highest point, and falls back to the ground. Assume that the forces acting on the shoe are gravity and drag due to air resistance. Assume that the drag force is proportional to and pointing in the opposite direction of the shoe s velocity; call the proportionality constant α. a 3 points Find an ODE for the shoe s velocity vt. b 4 points Solve your ODE, assuming the shoe s initial velocity is v 0. c 3 points If α = 3 gram/second and the shoe s mass is 200 grams, what should v 0 be so that the shoe reaches its maximum height after three seconds? 9 of 24
20 2. Consider the heat conduction problem u t = 4 u xx for x 0, 20 and t > 0 u x 0, t = 0, u20, t = 6 for t > 0 ux, 0 = e x for x [0, 20] a 4 points Find the steady-state solution vx. b 5 points The solution ux, t of the above heat conduction problem can be written as the sum of the steady-state solution and the transient solution: ux, t = vx + wx, t. Write down the heat conduction problem that the transient solution, wx, t, satisfies. 20 of 24
21 3. Consider the wave problem u tt = 9 u xx for x 0, π, t > 0 u0, t = uπ, t = 0 for t > 0 ux, 0 = 4 sin2x for x [0, π] u t x, 0 = 24 sin2x for x [0, π] a 5 points Find the solution. You may simply use your general solution from problem 9, if you wish. b 5 points Consider uπ/4, t: the displacement at the point x = π/4. This is a function of time. What is its minimum value? Find the sequence of times when this minimum value is achieved. 2 of 24
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24 If you ran out of room on any problem, write its continuation here! Make sure to label your work clearly. 24 of 24
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