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f(t)= @t, 0 ≦t < 1 1,1 ≤t <8

f(t)= @t, 0 ≦t < 1 1,1 ≤t <8

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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. (1², 1. f(1) = 2+1, 0≤ t ≤1 1<t≤2 2<t≤3 6-1, (1², 2. f(t) = (t-1)-¹, 3. f(t) = 1, meldets 1², 0 ≤t≤1 1<t<2 1, 3-t, 2 < t ≤3 0 ≤t≤1 1<t≤2 6. f(t) = cosh(bt) 7. f(t) = sinh(bt) Recall that 2 < t ≤3 4. Find the Laplace transform of each of the following functions: a. f(t) = t b. f(t) = 1² c. f(t) = t", where n is a positive integer 5. Find the Laplace transform of f(t) = cos(at), where a is a real constant. Recall that cosh(bt) = ½(eb¹ + e-b¹) and sinh(bt) = ½(ehr – e-bt). - 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. 8. f(t) = sin(bt) A. f(t) = cos(bt) f(t) = 1² sin(at) = In each of Problems 16 through 18, find the Laplace transform of the given function. 16. f(t) = 10. f(t) = eat sin(bt) 1. f(t) = eat cos(bt) 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. 365 12. f(t) = teat -13. f(t) = t sin(at) 14. f(1) = theat 15. 17. f(t) = ={{ 20. 1, 0≤t< T 0, to normido. 2≤1<∞0 In each of Problems 19 through 21, determine whether the given integral converges or diverges. worl.wroo 19. (1²+1)-¹dt 21. 18. f(t)=2-t, 0, 1, t, foc S To π≤1<∞ 0 ≤ t < 1 1≤t<∞ te ¹dt fredi t-² e' dt 0 ≤ t < 1 1≤t <2 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 §18 under less restrictive conditions, such as those of Theorem 6.1.2. 23. The Gamma Function. The gamma function is denoted by T(p) and is defined by the integral (7) apler. The integral converges as x → ∞ for all p. For p < 0 it is also 1 cos(bt) = (eibt + e-ibt) and sin(bt) di bien sw improper at x = 0, because the integrand becomes unbounded as (eibt - e-ibt). x → 0. However, the integral can be shown to converge at x = 0 2i for p > -1. a. 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. dog asd T(p+1) = Show that, for p > 0, 1 ex. I(p+1) = pr(p). e-xxPdx. b. Show that I (1) = 1. c. If p is a positive integer n, show that T(n + 1) = n!. Since I'(p) 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, p(p+1)(p+2)(p+n-1) = possible to show that I' Thus I (p) can be determined for all positive values of p if I (p) is known in a single interval of unit length-say, 0 < p ≤ 1. It is 3 2 () 2 = T(p+n) T(p) √. Find I (1/1). 2 and r IT
Transcribed Image Text: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. (1², 1. f(1) = 2+1, 0≤ t ≤1 1<t≤2 2<t≤3 6-1, (1², 2. f(t) = (t-1)-¹, 3. f(t) = 1, meldets 1², 0 ≤t≤1 1<t<2 1, 3-t, 2 < t ≤3 0 ≤t≤1 1<t≤2 6. f(t) = cosh(bt) 7. f(t) = sinh(bt) Recall that 2 < t ≤3 4. Find the Laplace transform of each of the following functions: a. f(t) = t b. f(t) = 1² c. f(t) = t", where n is a positive integer 5. Find the Laplace transform of f(t) = cos(at), where a is a real constant. Recall that cosh(bt) = ½(eb¹ + e-b¹) and sinh(bt) = ½(ehr – e-bt). - 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. 8. f(t) = sin(bt) A. f(t) = cos(bt) f(t) = 1² sin(at) = In each of Problems 16 through 18, find the Laplace transform of the given function. 16. f(t) = 10. f(t) = eat sin(bt) 1. f(t) = eat cos(bt) 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. 365 12. f(t) = teat -13. f(t) = t sin(at) 14. f(1) = theat 15. 17. f(t) = ={{ 20. 1, 0≤t< T 0, to normido. 2≤1<∞0 In each of Problems 19 through 21, determine whether the given integral converges or diverges. worl.wroo 19. (1²+1)-¹dt 21. 18. f(t)=2-t, 0, 1, t, foc S To π≤1<∞ 0 ≤ t < 1 1≤t<∞ te ¹dt fredi t-² e' dt 0 ≤ t < 1 1≤t <2 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 §18 under less restrictive conditions, such as those of Theorem 6.1.2. 23. The Gamma Function. The gamma function is denoted by T(p) and is defined by the integral (7) apler. The integral converges as x → ∞ for all p. For p < 0 it is also 1 cos(bt) = (eibt + e-ibt) and sin(bt) di bien sw improper at x = 0, because the integrand becomes unbounded as (eibt - e-ibt). x → 0. However, the integral can be shown to converge at x = 0 2i for p > -1. a. 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. dog asd T(p+1) = Show that, for p > 0, 1 ex. I(p+1) = pr(p). e-xxPdx. b. Show that I (1) = 1. c. If p is a positive integer n, show that T(n + 1) = n!. Since I'(p) 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, p(p+1)(p+2)(p+n-1) = possible to show that I' Thus I (p) can be determined for all positive values of p if I (p) is known in a single interval of unit length-say, 0 < p ≤ 1. It is 3 2 () 2 = T(p+n) T(p) √. Find I (1/1). 2 and r IT
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