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PSU MATH 251 - MATH 251 EXAM

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MATH 2512nd examNov. 12, 2002Name:Student Number:Instructor:Section:There are 9 partial credit questions. In order to obtain full credit for these prob-lems, all work must be shown. Credit will not be given for an answer not sup-ported by work.THE USE OF CALCULATORS IS NOT PERMITTED IN THIS EXAMINA-TION. At the end of the examination, the booklet will be collected.MATH 251 -2nd exam-1. Transform the given initial value problem into an initial value problem for two first orderequations.x2y00+ xy0+ (x2− λ2)y = 0, y(0) = y0, y0(0) = y002. What is the form of the particular solution to the equationy00+ 4y = e2t+ sin(2t) − 3 cos(t) + t2− 4DO NOT SOLVE FOR THE CONSTANTS!Page 2 of 8MATH 251 -2nd exam-3. (a) Sketch the function.f(t) = t − u1(t)(t − 1) + u3(t)(2 − t)(b) Write the given function in terms of unit step functions.g(t) =t20 ≤ t < 26 − t 2 ≤ t < 60 6 ≤ tPage 3 of 8MATH 251 -2nd exam-4. A mass of 3 kg is attached to a spring with spring constant k = 12Nm. What value ofthe damping coefficient γ will make the system critically damped?5. Find the Laplace transform.L{(eπtsin(3t)000}HINT:(eπtsin(3t)0= πeπtsin(3t) + 3eπtcos(3t)(eπtsin(3t)00= (π2+ 9)eπtsin(3t) + 6πeπtcos(3t)Page 4 of 8MATH 251 -2nd exam-6. Find the solution to the following initial value problem.y00+ 2y0− 8y = e3tu3(t), y(0) = 0, y0(0) = 1Page 5 of 8MATH 251 -2nd exam-7. Solve the initial value problem.y00+ 5y0+ 4y = 2δ(t − 3), y(0) = 1, y(0) = 0Page 6 of 8MATH 251 -2nd exam-8. Find the general solution. ~x0=1 43 0~x, ~x =70.Page 7 of 8MATH 251 -2nd exam-9. Find the general solution to the following systems of equations.(a) ~x0=1 −33 1~x(b) ~x0=3 2−2 −1~xPage 8 of


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PSU MATH 251 - MATH 251 EXAM

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