Lista de Exercícios Resolvidos - Transformada de Laplace - Boyce e DiPrima

1 de 1 Lista 7 Transformada de Laplace extraídos do livro de Boyce e DiPrima Problems In each of Problems 1 through 3 sketch the graph of the given function In each case determine whether f is continuous piecewise continuous or neither on the interval 0 t 3 1 ft t2 0 t 1 2 t 1 t 2 6 t 2 t 3 2 ft t2 0 t 1 t 1 1 1 t 2 1 2 t 3 3 ft t2 0 t 1 1 1 t 2 3 t 2 t 3 4 Find the Laplace transform of each of the following functions a ft e2 b ft t2 c ft t n where n is a positive integer 5 Find the Laplace transform of f t cosat where a is a real constant Recall that cosh br 12 ebr ebr and sinh br 12 ebr ebr In each of Problems 6 through 7 use the linearity of the Laplace transform to find the Laplace transform of the given function a and b are real constants 6 ft coshbr 7 ft sinhbr Recall that cosh br 12 ebr ebr and sinh br 12 ebr ebr In each of Problems 8 through 11 use the linearity of the Laplace transform to find the Laplace transform of the given function a and b are real constants Assume that the necessary elementary integration formulas extend to this case 8 ft sinbr 9 ft cosbr 10 ft c n sinbr 11 ft c n cosbr In each of Problems 12 through 15 use integration by parts to find the Laplace transform of the given function n is a positive integer and a is a real constant 12 ft t e t 13 ft t sinat 14 ft t 2 e t 15 ft t 2 sinat In each of Problems 16 through 18 find the Laplace transform of the given function 16 ft 1 0 t π 0 π t 17 ft 0 t 1 1 1 t 18 ft 2 t 0 t 1 1 t 2 0 2 t In each of Problems 19 through 21 determine whether the given integral converges or diverges 19 0 t2 1 1 dt 20 0 t e t dt 21 1 r 2 e t dt 22 Suppose that f and f are continuous for t 0 and of exponential order as t Use integration by parts to show that if F s L f t then lim F s 0 The result is actually true under less restrictive conditions such as those of Theorem 612 23 The Gamma Function The gamma function is denoted by Ip and is defined by the integral Ιp 1 0 e x x p d x 7 The integral converges as x for all p For p 0 it is also improper at x 0 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 for p 0 Γp 1 pΓp b Show that I1 1 c If p is a positive integer n show that In 1 n Since Lp is also defined when p is not an integer this function provides an extension of the factorial function to nonintegral values of the independent variable Note that it is also consistent to define 0 1 d Show that for p 0 pp 1p 2 p n 1 Γp n Γp Thus Γ p can be determined for all positive values of p if Γp is known in a single interval of unit length say 0 p 1 It is possible to show that Γ 12 π Find Γ 32 and Γ 11 2