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NAME: ID.NO: Problem Score Problem I.(24) VI.(8) II.(10) VII.(18) Score III.(12) IV.(12) V.(16) Total MA 266, Third MidTerm Examination, Wilkerson Section Second Part, April 17, 2001 GRIS 180 7PM You must have completed and turned in the first portion of the exam before you can receive and work on this second part. Do all your work on the question sheets. Calculators are NOT allowed. NO BOOKS OR PAPERS ARE ALLOWED. Use the back of the test pages for scrap paper.There is a short table of LaPlace transforms on the last page. You can remove this for use. I. (24 points) Already turned in. II. For the following problem, use Laplace transform method to solve the initial value problem: (10 points ) y 00 − y 0 = et , y(0) = 1, y 0 (0) = 0 Stop after finding the transform of the solution. 1 III.Find a particular solution to following non-homogeneous problem using your choice of method (12 points) y 00 − 4y 0 + 4y = te2t IV.(12 points) A spring mass problem with friction is abstracted to the second order non-homogeneous DE y 00 + 4y 0 + 9y = cos(3t) If the initial data y(0) = 1, y 0 (0) = 0, a) find the solution. b) what is the dominant term as t → ∞ ? c) What is the largest amplitude the solution makes? Use your choice of methods. 2 V.(6 points each) (a) L {4t + e−2t sin 3t} = (b) L {u2 (t)(e2t + e−2t )} = VI. Use the step functions {uc (t)} to solve. (8 points) (a) f (t) = 0 if t < 1 f (t) = 3 if 1 ≤ t < 2 f (t) = 1 if t ≥ 2 Sketch the graph of f (t) and find L {f (t)} = 3 VII.(6 points each) (a) If y 00 + y = δ(t − 2) + u3 (t), y(0) = 1, y 0 (0) = 21, then find (b)L {∫ t } −5τ sin(t − τ ) dτ = 0 e (c) L−1 {(s)/(s2 − 2s − 8)} = Hint: use partial fractions. 4 L {y} . Do not solve for y(t).