Solutions to Math 324 Homework 4: Finding Laplace Transforms of Specific Functions - Prof., Assignments of Differential Equations

Download Solutions to Math 324 Homework 4: Finding Laplace Transforms of Specific Functions - Prof. and more Assignments Differential Equations in PDF only on Docsity! Math 324 Homework 4 Solutions Instructor: J. Metcalfe Section 7.5, # 18 Find the Laplace transforms of the function: f(t) = cos 1 2 πt if 3 ≤ t ≤ 5; f(t) = 0 otherwise We begin by noting that f(t) = (u3(t)− u5(t))cos12πt. We apply the Laplace trans- form to this function, and using the cosine sum formula, obtain the following: L{f(t)} = L { u3(t)cos 1 2 πt } − L { u5(t)cos 1 2 πt } = L { u3(t)cos 1 2 π(t− 3 + 3) } − L { u5(t)cos 1 2 π(t− 5 + 5) } = L { u3(t)cos 1 2 π(t− 3)cos3π 2 − sin1 2 π(t− 3)sin3π 2 } − L { u5(t)cos 1 2 π(t− 5)cos5π 2 − sin1 2 π(t− 5)sin5π 2 } = L { u3(t)sin 1 2 π(t− 3) } + L { u5(t)sin 1 2 π(t− 5) } = e−3s π/2 s2 + π2/4 + e−5s π/2 s2 + π2/4 = 2π 4s2 + π2 (e−3s + e−5s) Section 7.5, # 29 Suppose that f(t) is a half-wave rectification of sin kt (Fig 7.5.16 in your book). Show that: L{f(t)} = k (s2 + k2)(1− e−πs/k) . We use the formula from Theorem 2 in your book to obtain: 1