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Unformatted text preview: MthSc 208, Spring 2011 (Differential Equations) Dr. Macauley HW 14 Due Thursday, March 31st, 2011 (1) Find the inverse Laplace transform of the following functions. 2 (a) Y (s) = 3  5s 1 (b) Y (s) = 2 s +4 5s (c) Y (s) = 2 s +9 3 (d) Y (s) = 2 s 3s + 2 (e) Y (s) = 2 s + 25 2  5s (f) Y (s) = 2 s +9 s (g) Y (s) = (s + 2)2 + 4 3s + 2 (h) Y (s) = 2 s + 4s + 29 2s  2 (i) Y (s) = (s  4)(s + 2) 3s2 + s + 1 (j) Y (s) = (s  2)(s2 + 1) (2) Use the Laplace transform to solve the following initial value problems. (a) y  4y = e2t t2 , y(0) = 1 (b) y  9y = 2et , y(0) = 0, y (0) = 1 (3) Find the Laplace transform of the given functions. (a) 3H(t  2) (b) (t  2)H(t  2) (c) e2(t1) H(t  1) (d) H(t  /4) sin 3(t  /4) (e) t2 H(t  1) (f) et H(t  2) (4) In this exercise, you will examine the effect of shifts in the time domain on the Laplace transform (graphically). (a) Sketch the graph of f (t) = sin t in the time domain. Find the Laplace transform F (s) = L{f (t)}(s). Sketch the graph of F in the sdomain on the interval [0, 2]. (b) Sketch the graph of g(t) = H(t  1) sin(t  1) in the time domain. Find the Laplace transform G(s) = L{g(t)}(s). Sketch the graph of G in the sdomain on the interval [0, 2] on the same axes used to sketch the graph of F . (c) Repeat the directions in part (b) for g(t) = H(t  2) sin(t  2). Explain why engineers like to say that "a shift in the time domain leads to an attenuation (scaling) in the sdomain." (5) Use the Heaviside function to concisely write each piecewise function. 5 2 t < 4; (a) f (t) = 0 otherwise t < 0; 0 t 0t<3 (b) f (t) = 4 t3 t < 0; 0 t2 0t<2 (c) f (t) = 4 t2 2 (6) Find the inverse Laplace transform of each function. Create a piecewise definition for your solution that doesn't use the Heavyside function. e2s (a) F (s) = s+3 1  es (b) F (s) = s2 es (c) F (s) = 2 s +4 ...
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This note was uploaded on 03/11/2012 for the course MTHSC 208 taught by Professor Staufeneger during the Spring '09 term at Clemson.
 Spring '09
 Staufeneger
 Differential Equations, Equations

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