s11_mthsc208_hw14

s11_mthsc208_hw14 - MthSc 208, Spring 2011 (Differential...

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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 = e-2t 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(t-1) H(t - 1) (d) H(t - /4) sin 3(t - /4) (e) t2 H(t - 1) (f) e-t 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 s-domain 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 s-domain 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 s-domain." (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. e-2s (a) F (s) = s+3 1 - e-s (b) F (s) = s2 e-s (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.

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s11_mthsc208_hw14 - MthSc 208, Spring 2011 (Differential...

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