AMATH 351 Autumn 2014 Homework 5 Due: Wednesday, November 5, 2014, 10:30AM Show work for full credit! 1. Find the Laplace transform of the function. s > |b| (a) f (t) = cosh bt, (b) f (t) = eat cos bt, s>a 2. Laplace transform I Use the Laplace transform to solve the initial value problem. (a) y 00 + 3y 0 + 2y = 0, y(0) = 1, y 0 (0) = 0 (b) y 00 − 4y 0 + 4y = 0, y(0) = 1, y 0 (0) = 1 (c) y 00 + 2y 0 + 5y = 0, y(0) = 2, y 0 (0) = −1 (d) y 00 + 4y = 4t, y(0) = 1, y 0 (0) = 5 3. The Heaviside function Sketch a graph of the given functions on the interval t ≥ 0. uc (t) represents the Heaviside (unit step) function. (a) g(t) = u1 (t) + 2u3 (t) − 6u4 (t) (b) g(t) = sin(t − 3)u3 (t) Express f (t) in (c) f (t) = terms of uc (t) and sketch a graph of f (t). t, 2, 7 − t, 0, 0≤t<2 2≤t<5 5≤t<7 t≥7 1 4. The Gamma function The Gamma function is denoted by Γ(p) and is defined by the integral: Z ∞ Γ(p + 1) = e−x xp dx 0 The integral converges as x → ∞ for all p. For p < 0 is it improper because the integrand becomes unbounded as x → 0. However, the integral can be shown to converge at x = 0 for p > −1. (a) Show that Γ(p + 1) = pΓ(p) for p > 0. (b) Show that Γ(1) = 1. (c) Using parts (a) and (b), find Γ(p) for p = 2, 3, 4, 5, 6. Is there a pattern? Find the pattern and write the formula for Γ(p) for any integer p > 0. Since Γ(p) is also defined when p is not an integer, this function provides an extension of the factorial function √ to non-integer values of the independent variable. It is possible to show that Γ(1/2) = π. (d) Find Γ(3/2) and Γ(11/2). 2

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