TESTETESTEINDEXAÇÃ0 1726751312828 Lista 7- Transformada de Laplace pdf

Lista 7: Transformada de Laplace (extraidos do livro de Boyce e DiPrima) Problems In each of Problems | through 3, sketch the graph of the given In each of Problems 16 through 18, find the Laplace transform of the function. In each case determine whether f is continuous, piecewise given function. continuous, or neither on the interval 0 < t < 3. 16 () = 1, O<t<7 1, 0<r<1 » fO= 0, 7 <t<o lL. f()=<24+t, 1<t<2 t, O<t<l 6-t, 2<t<3 17. fy= 1, L<t<oo t?, O<t<l t O<t<1 2 f= ((t-D)4, l<r<2 18. f(t) =42-1t, 1<t<2 1, 2<t<3 0, 2<t<o r, O<st<l In each of Problems 19 through 21, determine whether the given 3. fM=4h l<rs2 integral converges or diverges. 3-t, 2<t<3 oo 2 -1 4. Find the Laplace transform of each of the following functions: 19. [ (e+ Dd a. f(t)=t 00 b. f(y =? 20. | te dt c. f(t) =t", where n is a positive integer Oo 5. Find the Laplace transform of f(t) = cos(at), where a is a real 21. / te'dt constant. | . 22. Suppose that f and f’ are continuous for tf > 0 and of Recall that exponential order as tf — oo. Use integration by parts to show that cosh(bt) = b(e% +e") and sinh(bt) = be — ey, if F(s) = L{f()}, then lim F(s) = 0. The result is actually true 2 2 s—>00 In each of Problems 6 through 7, use the linearity of the Laplace under less restrictive conditions, such as those of Theorem 6.1.2. transform to find the Laplace transform of the given function; a and b 23. The Gamma Function. The gamma function is denoted by are real constants. I(p) and is defined by the integral oO 6. f(t) =cosh(bt) (p+) = [ et xP dx. (7) 7. f(t) = sinh(br) 0 Recall that The integral converges as x — ov for all p. For p < 0 it is also l b l b improper at x = 0, because the integrand becomes unbounded as cos(bt) = ic "+e ') and sin(bt) = ye ae), x — 0. However, the integral can be shown to converge at x = 0 f >-i. In each of Problems 8 through 11, use the linearity of the Laplace orp transform to find the Laplace transform of the given function; a and b a. Show that, for p > 0, are real constants. Assume that the necessary elementary integration P(p +1) = pl(p). formulas extend to this case. b. Show thatl'(1) = 1. 8. f(t) = sin(bt) c. If pis a positive integer n, show that 9. f(t) =cos(bt) T(n+1) =a. 10. f(t) =e sin(bt) Since I’( p) is also defined when p is not an integer, this function _ wat provides an extension of the factorial function to nonintegral He f(t) = eM cost br) values of the independent variable. Note that it is also consistent In each of Problems 12 through 15, use integration by parts to find the to define 0! = 1. Laplace transform of the given function; n is a positive integer and a d. Show that, for p > 0, is a real constant. (pt D(p+2)-(p4n—l) = T(p +n) 12. f(t) =te” PAP P P ~ Tp) * 13. f(t) =tsin(at) Thus I’\( p) can be determined for all positive values of p if I'( p) 14. f(t) =r"e" is known in a single interval of unit length—say,0 < p < 1. Itis . 1 3 11 15. f(t) =f sin(at) possible to show that I (5) = /m.FindD (5) andl (5) .