Lecture Notes in Differential Equations - Bruce E. Shapiro

335
Example 32.4. From integral A.107,
∫ ∞
L[cos kt] = cos kt e −st dt
0
∣
= e−st (k sin kt − s cos kt) ∣∣∣
∞
k 2 + s 2
0
s
=
s 2 + k 2
Example 32.5. From integral A.105,
Example 32.6.
L [ e at] =
∫ ∞
0
∫ ∞
L[sin kt] = sin kt e −st dt
0
∣
= e−st (−s sin kt − k cos kt) ∣∣∣
∞
k 2 + s 2
0
k
=
s 2 + k 2
e at e −st dt = e(a−s)t
a − s
Example 32.7. Using linearity and example 32.6,
]
1
L[cosh at]dt = L[
2 (eat + e −at )
Example 32.8. Using linearity and example 32.6,
]
1
L[sinh at]dt = L[
2 (eat − e −at )
∣
∞
0
= 1
s − a , s > a (32.19)
(32.20)
= 1 ( [
L e
at ] + L [ e −at]) (32.21)
2
= 1 ( 1
2 s − a + 1 )
(32.22)
s + a
s
=
s 2 − a 2 (32.23)
(32.24)
= 1 ( [
L e
at ] − L [ e −at]) (32.25)
2
= 1 ( 1
2 s − a − 1 )
(32.26)
s + a
a
=
s 2 − a 2 (32.27)