Midterm Test Nov 10, 2010 Student Number:

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1 Mathematic 265 Section: 03 Verion A Full Name: Midterm Tet Nov 0, 200 Student Number: Intruction: There are 6 page in thi tet (including thi cover page).. Caution: There may (or may not) be more than one verion of thi tet paper. 2. Enure that your full name and tudent number appear on thi page. Circle your ection number. 3. No calculator, book, note, or electronic device of any kind are permitted. 4. Show all your work. Anwer not upported by calculation or reaoning will not receive credit. Mey work will not be graded. 5. Five minute before the end of the tet period you will be given a verbal notice. After that time, you mut remain eated until all tet paper have been collected. 6. When the tet period i over, you will be intructed to put away writing implement. Put away all pen and pencil at thi point. Continuing to write pat thi intruction will be conidered dihonet behaviour. 7. Pleae remain eated and pa your tet paper down the row to the nearet indicated aile. Once all the tet paper have been collected, you are free to leave. 8. Expoing your tet paper, copying from another tudent paper, or haring information about thi tet contitute academic dihonety. Such behaviour may jeopardize your grade on thi tet, in thi coure, and your tanding at thi univerity. 9. There i a table of Laplace Tranform on p 6 of thi tet paper. Quetion number Grade Value I have read and undertood the intruction and agree to abide by them. Signed: 4 6 Total 60

2 Problem : Multiple Choice Quetion: Circle ONE correct anwer (a, b, c, d, or e). There i no partial credit in thi quetion. Illegible or multiple anwer will get no credit. NOTE: In thee quetion, the notation for tep function include u c (t) = H(t c). : The Laplace tranform of the function f(t) = t 2 i (a) e 2, (b) e 2, (c) u 2 ()e 2, (d), (e) None of thee. 2 Which of the below correpond to the invere Laplace tranform of F() = 3 e (a) H(t )e t, (b) H(t 3)e t, (c) H(t 3)e t, (d) H(t )e 3(t ), (e) H(t 3)e t 3 3 For the differential equation y + 2y + y = δ(t) with initial condition y(0) = 0, y (0) = 0, we find that F() = L{y(t)} i which of the following function? (a) t 0 y(t τ)δ(τ)dτ, (b) , (c) , (d) e , (e) ( + ) 2 4: A pirate-hip ride at an amuement park i driven by a motor o that the vertical diplacement of the hip atifie αz + βz + γ z = 0 co(ωt), where α, β, γ are manufacturer pecification for the ride, and ω i the driving frequency of the motor powering the ride. Suppoe that α = 28, γ = 7 are fixed. For which of the following etting will the diplacement of the ride have the greatet amplitude? (a) β = 0.05, ω = /4, (b) β = 0.05, ω = 4, (c) β = 2, ω = 3/28, (d) β = 2, ω = /2, (e) β = 4, ω = /4 5: Conider the ytem of firt order ODE given below. dx dt dy dt = x 2y, = x + 3y. Which of thee anwer bet decribed the behaviour of the ytem? (a) Growing exponential behaviour (b) Decaying exponential behaviour (c) Combined growing and decaying exponential (d) Ocillation with growing amplitude (e) Ocillation with decaying amplitude 2

3 6: Conider the differential equation y + 6y = E 0 co(ωt). The following behaviour i oberved when the driving frequency i ω = 4.4. What i the frequency (radian/time) of the envelope of the ocillation and of the ocillation themelve? (a) Envelope frequency= 8.4, ocillation frequency 0.4 (b) Envelope frequency= 0.4, ocillation frequency 8.4 (c) Envelope frequency= 6/4.4, ocillation frequency (d) Envelope frequency= 0.2, ocillation frequency 4.2 (e) Envelope frequency= 4.2, ocillation frequency 0.2 Problem 2: Conider the function hown in the figure. Anwer (i) and (ii). (i) Find the Laplace Tranform of thi function. L{f(t)} = (ii) If the ame function i extended o that it i periodic, with period T = 5, what would be the Laplace tranform of thi new function? L{f p (t)} = 3

4 Problem 3: Short Anwer Quetion Check your anwer carefully, a there are no part mark for right method() in thi quetion. (A) Find the invere Laplace tranform of the following function: (a) F() = 9 5 L {F()} = (b) F() = 2 ( + 2)( + ) L {F()} = (c) F() = ( ) L {F()} = (B) Conider the ODE y + 9y = t co(3t) Uing the Method of Undetermined Coefficient, what would be the form of the particular olution to thi equation? NOTE: Do not olve the equation and do not find the coefficient. y p (t) = 4

5 Problem 4: In a frictionle pring-ma ytem, the ma i m = 2kg and the pring contant i k = 2m kg 2. The vertical diplacement of the pring atifie m d2 y dt 2 + ky = F(t), y(0) = 0, y (0) = 0. () (a) At t = 0, the ma i hit with a hammer, producing a unit impule. You may aume that thi impule i repreented by the Dirac delta function. Solve Eqn. () to find the diplacement y(t) at t > 0. y(t) = (b) What would be the velocity of the ma, v(t) in the ame ytem with the ame applied force? v(t) = (c) Conider the function g(t) = t and h(t) = in(t). Compute the convolution g h. [Hint: A ueful fact i: xin(x)dx = in(x) xco(x).] g h = Problem 4 i continued next page.. 5

6 Problem 4 Cont d (d) In a new experiment, a time dependent force of the form g(t) = t i applied to the pring-ma ytem. Ue the Laplace tranform method to olve Eqn. () with thi time-dependent force. (You may find the reult in (c) ueful.) f(t) F() = L[f(t)] e at a t n n! n+ in at a 2 +a 2 co at 2 +a 2 u c (t) e c u c (t)f(t c) e c F() δ(t c) e c e ct f(t) F( c) t f(t τ)g(τ) dτ F()G() 0 6