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Unformatted text preview: MthSc 208, Spring 2011 (Differential Equations) Dr. Macauley HW 13 Due Friday March 18th, 2011 (1) Solve the following differential equations: (a) y + 6y + 9y = 5 (b) y = - 2 y (c) y + 2y = et (d) y + 3y = 0. (2) Find the Laplace transform of the following functions by explicitly computing
0 f (t) e-st dt. (a) f (t) = 3 (b) f (t) = e3t (c) f (t) = cos 2t (d) f (t) = te2t (e) f (t) = e-3t sin 2t (3) Sketch each of the following piecewise defined functions, and compute their Laplace transforms. 0, 0 t < 4 t, 0 t < 3 (a) f (t) = (b) f (t) = 5, t 4 3, t 3 (4) Engineers frequently use the Heavyside function, defined by H(t) = 0, 1, t<0 t0 to emulate turning on a switch at a certain instance in time. Sketch the graph of the function x(t) = e0.2t and compute its Laplace transform, X(s). On a different set of axes, sketch the graph of y(t) = H(t - 3)e0.2t and calculate its Laplace transform, Y (s). How do X(s) and Y (s) differ? What do you think the Laplace transform of H(t - c)e0.2t is, where c is an arbitrary positive constant? (5) Find the Laplace transform of the following functions by using a table of Laplace transforms (a) f (t) = -2 (b) f (t) = e-2t (c) f (t) = sin 3t (d) f (t) = te-3t (e) f (t) = e2t cos 2t (6) Transform the given initial value problem into an algebraic equation involving Y (s) := L(y), and solve for Y (s). (a) y + y = sin 4t, y(0) = 0, y (0) = 1 (b) y + y + 2y = cos 2t + sin 3t, y(0) = -1, y (0) = 1 (c) y + y = e-t sin 3t, y(0) = 0 ...
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