Calculate the Laplace transform Lleft{sin(t-k) cdot H(t-k)right}

Tazmin Horton 2020-11-01 Answered
Calculate the Laplace transform
L{sin(tk)H(tk)}
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Expert Answer

Aubree Mcintyre
Answered 2020-11-02 Author has 73 answers

Step 1
To find the Laplace transform of
L{sin(tk)H(tk)}
Step 2
Since,
For the Heaviside function (or unit step function)
(Ha(t)=H(ta)={0ta)The second shifting theorem,
If Lf(t)=F(s) then LH(tk)f(ta)=easF(s)
First we need to find the laplace transform of sin(t).
Step 3
Let ,f(t)=sin(t)
L{f(t)}=Lsin(t)
Since ,L{f(t)}=a(s2+a2)
L{f(t)}=1(s2+1)
Thus ,F(s)=1(s2+1)
Step 4
By second shifting theorem,
If Lf(t)=F(s)then LH(tk)f(ta)=easF(s)
Thus,
L{H(tk)f(ta)}=eksF(s)
=eks(1(s2+1))
=eks(s2+1)
Step 5
Therefore, L{sin(tk)H(tk)}=eks(s2+1)

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