2^nd Translation Property of Laplace Transform

1. MD. MUSFIQUR RAHMAN FOYSAL
ID: 163-15-8489
2nd
Translation Property of Laplace Transform
Statement of 2nd
Translation is:
If ℒ 𝐹 𝑡 = 𝑓(𝑠) and 𝐺 𝑡 =
𝐹(𝑡 − 𝑎), 𝑡 > 𝑎
0, 𝑡 < 𝑎
then, ℒ{𝐺(𝑡)} = 𝑒−𝑎𝑠
𝑓(𝑠)

2. Soln: Let F(t)=Sin2t
=
1
2
. 2Sin2t =
1
2
(1-cos2t)
=> ℒ{F(t)} =
1
2
ℒ{1-cos2t}
=> f(s) =
1
2
1
2
−
𝑠
𝑠2+4
=
2
𝑠(𝑠2+4)
=> ℒ{G(t)} = 𝑒−
𝜋
5
𝑠
f(s)
=
2𝑒
−
𝜋
5
𝑠
𝑠(𝑠2+4)
∴ This is the required solution.
Example: Evaluate: If 𝑮 𝒕 =
𝒔𝒊𝒏𝟐(𝒕 −
𝝅
𝟓
), 𝒕 >
𝝅
𝟓
𝟎, 𝒕 <
𝝅
𝟓
then, 𝓛 𝑮 𝒕 =?