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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 al is a real constant.

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 al is a real constant.

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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1. equation (6), we write Then, from Examples 5 and 7, we obtain L{f(t)} 2. f(t) = 6-t, (1², 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, f(t)=2+t, (t-1)-¹, 1, t², 1, = L{ f(t)} = 5£{e-²¹} - 3£{sin(41)}. 0 ≤ t ≤ 1 1< t <2 2<t≤3 cosh(bt) = (+ Va 0 ≤t≤1 1< t ≤2 2 < t ≤3 = 0≤t≤1 1<t≤2 2<t≤3 5 S+2 12. f(t) = teat 13. f(t) = t sin(at) 14. f(t) = teat 15. f(t) = 1² sin(at) - ul6V 12 s² + 16' s > 0. 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 In each of Problems 16 through 18, find the Laplace transform of the given function. In each of Problems 6 through 7, use the linearity of the Laplace lov transform to find the Laplace transform of the given function; a and b are real constants. 6. f(t) = cosh(bt) 7. f(t) = sinh(bt) Recall that 1 cos(bt) gibt + e-ibt) and sin(bt) = (eibt - e-ibt). žleibe 1 2i 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. 16. f(t)= 1≤t<∞0 0 ≤ t < 1 11 < 2 2≤t<∞0 to noituldo, In each of Problems 19 through 21, determine whether the given integral converges or diverges. Isitial to go 3. f(t) = 3-t, 7011 weroo 4. Find the Laplace transform of each of the following functions: no: 19. S (t² + 1)-¹dt 17. f(t) = 8. f(t) = sin(bt) 9. f(t) = cos(bt) 10. f(t) = eat sin(bt) 11. 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. 18. f(t) = worl = 20. 21. 22. Suppose that f and f' are continuous for t≥ 0 and of exponential order as t→∞. Use integration by parts to show that 1 (eb¹ + e−br) and sinh(bt) = -(e¹¹ - e-bt). be an if F(s) = £[ƒ(1)), then lim F(s) = 0. The result is actually true L{f(t)}, Sintienos no 12: 818 under less restrictive conditions, such as those of Theorem 6.1.2. 1, 0≤t<T 10, T ≤t<∞ [t, 0≤t<1 1, t, 2-t, 8 So t vd betonsb te tdt [₁ t-2 e' dt 23. The Gamma Function. The gamma function is denoted by T(p) and is defined by the integral (7) The integral converges as x→ ∞ for all p. For p < 0 it is also adı 15blanos sw 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, 88 = √²² T(p+1) = b. Show that IT (1) = 1. c. T(p+1) = p (p). e-xxPdx. If p is a positive integer n, show that T(n + 1) = n!. possible to show that I' 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) = Thus I (p) can be determined for all positive values of p if T (p) is known in a single interval of unit length-say, 0 < p ≤ 1. It is T(p+n) T(p) 3 (1) = √T. Find I (²) 2 and I 11 (1) 2
Transcribed Image Text:1. equation (6), we write Then, from Examples 5 and 7, we obtain L{f(t)} 2. f(t) = 6-t, (1², 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, f(t)=2+t, (t-1)-¹, 1, t², 1, = L{ f(t)} = 5£{e-²¹} - 3£{sin(41)}. 0 ≤ t ≤ 1 1< t <2 2<t≤3 cosh(bt) = (+ Va 0 ≤t≤1 1< t ≤2 2 < t ≤3 = 0≤t≤1 1<t≤2 2<t≤3 5 S+2 12. f(t) = teat 13. f(t) = t sin(at) 14. f(t) = teat 15. f(t) = 1² sin(at) - ul6V 12 s² + 16' s > 0. 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 In each of Problems 16 through 18, find the Laplace transform of the given function. In each of Problems 6 through 7, use the linearity of the Laplace lov transform to find the Laplace transform of the given function; a and b are real constants. 6. f(t) = cosh(bt) 7. f(t) = sinh(bt) Recall that 1 cos(bt) gibt + e-ibt) and sin(bt) = (eibt - e-ibt). žleibe 1 2i 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. 16. f(t)= 1≤t<∞0 0 ≤ t < 1 11 < 2 2≤t<∞0 to noituldo, In each of Problems 19 through 21, determine whether the given integral converges or diverges. Isitial to go 3. f(t) = 3-t, 7011 weroo 4. Find the Laplace transform of each of the following functions: no: 19. S (t² + 1)-¹dt 17. f(t) = 8. f(t) = sin(bt) 9. f(t) = cos(bt) 10. f(t) = eat sin(bt) 11. 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. 18. f(t) = worl = 20. 21. 22. Suppose that f and f' are continuous for t≥ 0 and of exponential order as t→∞. Use integration by parts to show that 1 (eb¹ + e−br) and sinh(bt) = -(e¹¹ - e-bt). be an if F(s) = £[ƒ(1)), then lim F(s) = 0. The result is actually true L{f(t)}, Sintienos no 12: 818 under less restrictive conditions, such as those of Theorem 6.1.2. 1, 0≤t<T 10, T ≤t<∞ [t, 0≤t<1 1, t, 2-t, 8 So t vd betonsb te tdt [₁ t-2 e' dt 23. The Gamma Function. The gamma function is denoted by T(p) and is defined by the integral (7) The integral converges as x→ ∞ for all p. For p < 0 it is also adı 15blanos sw 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, 88 = √²² T(p+1) = b. Show that IT (1) = 1. c. T(p+1) = p (p). e-xxPdx. If p is a positive integer n, show that T(n + 1) = n!. possible to show that I' 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) = Thus I (p) can be determined for all positive values of p if T (p) is known in a single interval of unit length-say, 0 < p ≤ 1. It is T(p+n) T(p) 3 (1) = √T. Find I (²) 2 and I 11 (1) 2
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