082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

21.4 Properties of the Laplace transform 631
21.4 PROPERTIESOFTHELAPLACETRANSFORM
There are some useful properties of the Laplace transform that can be exploited. They
allow us to find the Laplace transforms of more difficult functions. The properties we
shall examine are:
(1) linearity;
(2) shift theorems;
(3) final value theorem.
21.4.1 Linearity
Let f andgbe two functions oft and letkbe a constant which may benegative. Then
L{f +g}= L{f}+L{g}
L{kf}=kL{f}
The first property states that to find the Laplace transform of a sum of functions, we
simply sum the Laplace transforms of the individual functions. The second property
says that if we multiply a function by a constant k, then the corresponding transform
is also multiplied by k. Both of these properties follow directly from the definition of
the Laplace transform and linearity properties of integrals, and mean that the Laplace
transformisalinearoperator.UsingthelinearitypropertiesandTable21.1,wecanfind
the Laplace transformsof more complicated functions.
Example21.3 Find the Laplace transforms ofthe following functions:
(a) 3 +2t (b) 5t 2 −2e t
Solution (a) L{3 +2t} = L{3} + L{2t}
= 3L{1} +2L{t}
= 3 s + 2 s 2
(b) L{5t 2 −2e t }= L{5t 2 } + L{−2e t }
= 5L{t 2 } −2L{e t }
= 10
s − 2
3 s −1
With a littlepractice, someofthe intermediatesteps maybeexcluded.
Example21.4 Findthe Laplace transforms ofthe following:
(a) 5cos3t+2sin5t−6t 3
Solution (a) L{5cos3t +2sin5t −6t 3 } =
(b) −e −t + 1 (sint +cost)
2
5s
s 2 +9 + 10
s 2 +25 − 36
s 4