630 LAPLACE TRANSFORMS
=[
(0− 0 −0)−(
0 − 0 −2
s^3)]=2
s^3(provideds>0)(c) From equation (1),
L{coshat}=L{
1
2(eat+e−at)}
,from Chapter 5=1
2L{eat}+1
2L{e−at},equations (2) and (3)=1
2(
1
s−a)
+1
2(
1
s−(−a))from (iii) of Table 64.1=1
2[
1
s−a+1
s+a]=1
2[
(s+a)+(s−a)
(s−a)(s+a)]=s
s^2 −a^2(provideds>a)Problem 4. Determine the Laplace transforms
of: (a) sin^2 t (b) cosh^23 x.(a) Since cos 2t= 1 −2sin^2 tthensin^2 t=1
2(1−cos2t). Hence,L{sin^2 t}=L{
1
2(1−cos 2t)}=1
2L{ 1 }−1
2L{cos 2t}=1
2(
1
s)
−1
2(
s
s^2 + 22)from (i) and (v) of Table 64.1=(s^2 +4)−s^2
2 s(s^2 +4)=4
2 s(s^2 +4)=2
s(s^2 +4)(b) Since cosh 2x=2 cosh^2 x−1 thencosh^2 x=1
2(1+cosh 2x) from Chapter 5.Hence cosh^23 x=1
2(1+cosh 6x)ThusL{cosh^23 x}=L{
1
2(1+cosh 6x)}=1
2L{ 1 }+1
2L{cosh 6x}=1
2(
1
s)
+1
2(
s
s^2 − 62)=2 s^2 − 36
2 s(s^2 −36)=s^2 − 18
s(s^2 −36)Problem 5. Find the Laplace transform of
3 sin (ωt+α), whereωandαare constants.Using the compound angle formula for sin(A+B),
from Chapter 18, sin(ωt+α) may be expanded to
(sinωtcosα+cosωtsinα). Hence,L{3sin (ωt+α)}
=L{3(sinωtcosα+cosωtsinα)}=3 cosαL{sinωt}+3 sinαL{cosωt},sinceαis a constant=3 cosα(
ω
s^2 +ω^2)
+3 sinα(
s
s^2 +ω^2)from (iv) and (v) of Table 64.1=3
(s^2 +ω^2 )(ωcosα+ssinα)Now try the following exercise.Exercise 231 Further problems on an intro-
duction to Laplace transformsDetermine the Laplace transforms in Problems
1to9.- (a) 2t−3 (b) 5t^2 + 4 t− 3
[
(a)
2
s^2−3
s(b)10
s^3+4
s^2−3
s]- (a)
t^3
24− 3 t+2 (b)t^5
15− 2 t^4 +t^2
2
[
(a)1
4 s^4−3
s^2+2
s(b)8
s^6−48
s^5+1
s^3]