constant. Recall that (t) = cos(at), where a is a rea 1 cosh(bt) = (ebt + e-bt) and sinh(bt) = (ebt - e-bt). 1 he
constant. Recall that (t) = cos(at), where a is a rea 1 cosh(bt) = (ebt + e-bt) and sinh(bt) = (ebt - e-bt). 1 he
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Question
9
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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