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Unformatted text preview: MthSc 208, Fall 2011 (Differential Equations) Dr. Macauley HW 15 Due Monday April 4th, 2011 (1) For each initial value problem, sketch the forcing term, and then solve for y(t). Write your solution as a piecewise function (i.e., not using the Heavysie function). Recall that the function Hab (t) = H(t  a)  H(t  b) is the interval function. (a) y + 4y = H01 (t), y(0) = 0, y (0) = 0 (b) y + 4y = t H01 (t), y(0) = 0, y (0) = 0 (2) Define the function 1 p (t) = (Hp (t)  Hp+ (t)) . (a) Show that the Laplace transform of p (t) is given by 1  es . s (b) Use l'H^pital's rule to take the limit of the result in part (a) as 0. How does this o result agree with the fact that L{p (t)} = esp ? (3) Use a Laplace transform to solve the follosing initial value problem: L p (t) = esp y = p (t), y(0) = 0 How does your answer support what engineers like to say, that the "derivative of a unit step is a unit impulse"? (4) Define the function 0t<p 0, 1 (x  p), pt<p+ Hp (t) = 1, tp+ (a) Sketch the graph of Hp (t). (b) Without being too precise about things, we could argue that Hp (t) Hp (t) as 0, where Hp (t) = H(t  p). Sketch the graph of the derivative of Hp (t). (c) Compare your result in (b) with the graph of p (t). Argue that Hp (t) = p (t). (5) Solve the following initial value problems. (a) y + 4y = (t), y(0) = 0, y (0) = 0 (b) y  4y  5y = (t), y(0) = 0, y (0) = 0 ...
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 Spring '09
 Staufeneger
 Differential Equations, Equations

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