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PSU MATH 251 - MATH 251 Midterm Exam II

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MATH 251Midterm Exam IIJuly 20, 2006Name:Student Number:Instructor:Section:This exam has 10 questions for a total of 100 points. There are 5 partial credit questions. Inorder to obtain full credit for partial credit problems, all work must be shown. Creditwill not be given for an answer not supported by work.THE USE OF CALCULATORS IS NOT PERMITTED IN THIS EXAMINATION.At the end of the examination, the booklet will be collected.1:2:3:4:5:6:7:8:9:10:Total:Do not write in this box.MATH 251 Summer 2006 Exam II1. (8 points) Consider a mass-spring system described by the differential equation2u00+ γu0+ 8u = 2 sin(ωt).Answer the following questions.(a) When γ = 0 what is the system’s natural period, T ?(b) For what value(s) of γ will the system be critically damped?(c) If γ = 4, is the system underdamped or overdamped?(d) (True of false) Resonance would occur if γ = 0 and ω = 2.2. (5 points) Suppose y(t) is the solution of the second order linear initial value problemy00+ 4y = t, y(0) = 1, y0(0) = 0.What is the Laplace transform of y(t)?(a) Y (s) =1s2(s2+ 4)(b) Y (s) =1 − s2s2(s2+ 4)(c) Y (s) =s2+ 1s2(s2+ 4)(d) Y (s) =s3+ 1s2(s2+ 4)Page 2 of 9MATH 251 Summer 2006 Exam II3. (5 points) What is the inverse Laplace transform of3s + 4s2+ 2s + 5?(a) 3etcos 2t −72etsin 2t(b) 3etcos 2t + 2etsin 2t(c) 3e−tcos 2t +12e−tsin 2t(d) 3e−tcos 2t + e−tsin 2t4. (5 points) Let f(t) = 1 − u1(t) + u5(t) (t − 3) + u8(t) t2. Then f(7) =(a) 2(b) 4(c) 68(d) f is undefined at t = 7.Page 3 of 9MATH 251 Summer 2006 Exam II5. (5 points) Which of the following systems of first order linear equations is equivalent to thesecond order linear equationy00− 2y0+ 5y = 0?(a)x01= x1x02= 5x1− 2x2(b)x01= x1x02= −5x1+ 2x2(c)x01= x2x02= 5x1− 2x2(d)x01= x2x02= −5x1+ 2x2Page 4 of 9MATH 251 Summer 2006 Exam II6. (16 points)(a) (6 points) When solving the following nonhomogeneous equation using the method ofundetermined coefficients, what is a suitable form of the prticular solution of Y (t) to use?(DO NOT ATTEMPT TO SOLVE FOR THE COEFFICIENTS!)y00+ 2y0− 3y = t2et+ e3tcos 2t − 3t sin 3t.(b) (10 points) Find the general solution of the nonhomogeneous linear equationy00− 4y0+ 3y = 3t2− 1.Page 5 of 9MATH 251 Summer 2006 Exam II7. (16 points) A mass of 2 kg stretches a spring 2 m. The system has a damping constant of 4kg/s. The mass is pulled 4 m downward from its equilibrium position and released with zeroinitial velocity. You may take g = 10 m/s2as the gravitational constant.(a) (12 points) Set up and solve an initial value problem to find the system’s diplacementfunction u(t).(b) (2 points) What is the quasi-frequency of the system?(c) (2 points) What is limt→∞u(t)?Page 6 of 9MATH 251 Summer 2006 Exam II8. (12 points) Rewrite the following piecewise continuous function f (t) in terms of the unit-stepfunctions, and then find its Laplace transform.f(t) =0 0 ≤ t < 23t − 6, 2 ≤ t < 44e−3t, 4 ≤ t.Page 7 of 9MATH 251 Summer 2006 Exam II9. (16 points) Use the Laplace transform to solve the initial value problemy00+ 4y0+ 8y = δ(t − π), y(0) = 0, y0(0) = −1.No credit will be given if the Laplace transform is not used to solve this problem.Page 8 of 9MATH 251 Summer 2006 Exam II10. (12 points)(a) (10 points) Solve the initial value problemx0=−2 12 −3x, x(0) =41(b) (2 points) What is limt→∞|x(t)|?Page 9 of


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