DE75E68Ed01

# DE75E68Ed01 - b If 39 e-3 t 8 δ t-2 π represents an external force applied to a mechanical systsem brieﬂy describe the action of this force 4

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M427K EXAM 3B Your name: SPRING, 2010 Dr. Schurle Your UTEID: Show all your work on these pages. Be organized and neat. Your work should be your own; there should be no talking, reading notes, checking laptops, using cellphones, . . . . 1. (20 points) Either solve the following boundary value problem or else show that it has no solution. x 2 y 00 - 2 xy 0 + 2 y = 0 , y (1) = 7 , y (2) = 22

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YOUR SCORE: /100 2. Let g ( t ) = ( 3 0 t 2 t 2 - 4 t + 7 2 t . (a) (8 points) Sketch an accurate graph of g ( t ) on t 0. Be sure to put scales on your x - and y - axes. (b) (8 points) Use one or more step functions and polynomials to express g ( t ). (c) (8 points) Find the Laplace transform of g ( t ). You may use the table attached at the end of this exam.
3. (32 points) a) Solve the following initial value problem, where δ ( t ) is the Dirac delta function. y 00 + 4 y = 39 e - 3 t + 8 δ ( t - 2 π ) , y (0) = 5 , y 0 (0) = 12 SHOW ALL YOUR WORK AND SIMPLIFY YOUR ANSWER!

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Unformatted text preview: b) If 39 e-3 t + 8 δ ( t-2 π ) represents an external force applied to a mechanical systsem, brieﬂy describe the action of this force. 4. Deﬁne the function f ( x ) by f ( x ) = (-x-4 ≤ x ≤ 4 < x < 4 , f ( x + 8) = f ( x ) . (a) (6 points) Sketch an accurate graph of f ( x ) for three periods. Be sure to put scales on your x-and y-axes. (b) (18 points) Find and simplify formulas for the coeﬃcients of the Fourier series for the function f ( x ). Do NOT use a calculator! Write out the ﬁrst ﬁve nonzero terms of this Fourier series. You may ﬁnd the following integrals helpful. Z x sin( ax ) dx = sin ax a 2-x cos ax a + C ; Z x cos( ax ) dx = cos ax a 2 + x sin ax a + C...
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## This note was uploaded on 10/19/2010 for the course M 427K taught by Professor Fonken during the Spring '08 term at University of Texas at Austin.

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DE75E68Ed01 - b If 39 e-3 t 8 δ t-2 π represents an external force applied to a mechanical systsem brieﬂy describe the action of this force 4

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