For f(t)=e^{-t}-1 , find the Laplace transform of \{\frac{f(t)}{t}\}.For g(t)=e^{-t}-2 ,examine if the Laplace transform of \{\frac{g(t)}{t}\} exists

geduiwelh 2021-09-17 Answered

For f(t)=et1 , find the Laplace transform of {f(t)t}. Then , for g(t)=et2 , examine if the Laplace transform of {g(t)t} exists.

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Expert Answer

tabuordg
Answered 2021-09-18 Author has 99 answers

Step 1
Given that, f(t)=et1
Now , we know laplace transformation of f(t) is L(f(t)) is given by L{f(t)}=0f(t)estdt
Also we know , property of unilafenal Linear transformation is L[f(t)t]=0
F(w)dw : where F(w) is the Laplace transformation of f(t)
Now , Laplace transformation of f(t)=et1 is
F(s)=L{f(t)}=L{et1}=0(et1)estdt
=0(e(s+1)test)dt
=[e(s+1)t(s+1)+ests]0
=1s+11s.(since  e=0)
Now , linear transformation of {f(t)t} is
L{f(t)t}=L{et1t}=sF(w)dw
=s[1w+11w]dw
=[ln(w+1)ln(w)]s
=[ln(w+1w)]s
=ln(s+1s).limwln(w+1w)=0
=ln(ss+1)
But for g(t)=et2
G(s)=L{g(t)}=1s+12s . Simply as previous

Now , L{g(t)t}=L{et2t}=sG(w)dw

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