Existence of Laplace Transform Do the Laplace transforms for the following functions exist? Explain your answers. (You do not need to find the transforms , just show if they exist or not) a) f(t)=t^2sin(omega t) b) f(t)=e^{t^2}sin(omega t)

Reeves 2020-12-16 Answered
Existence of Laplace Transform
Do the Laplace transforms for the following functions exist? Explain your answers. (You do not need to find the transforms , just show if they exist or not)
a) f(t)=t2sin(ωt)
b) f(t)=et2sin(ωt)
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Expert Answer

Luvottoq
Answered 2020-12-17 Author has 95 answers

Step 1
To determine:
The existence of Laplace transforms of the functions:
a) f(t)=t2sin(ωt)
b) f(t)=et2sin(ωt)
Step 2
Definition of existence of Laplace transforms :
Let f be a piece – wise continuous function in [0,) and is of exponential order. The Laplace transformation F (s) of f exists for some s>c, where c is a real number depends on f.
Step 3
a)Since, the function f(t)=t2sin(ωt) can be expressed as an exponential order. Hence the Laplace transform exists for the function f(t)=t2sin(ωt)
Step 4
b)The f(t)=et2sin(ωt) cannot be expressed as an exponential order, since it grows too fast than the exponential function and bounded by et and hence, the integral doesn’t converge
it would at some point exceed the est term.
This is the reason for the Laplace transformation of this function is undefined.

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