Find the laplace transform of the following: a) t^2 sin kt b) tsin kt

pedzenekO 2021-03-05 Answered
Find the laplace transform of the following:
a)t2sinkt
b)tsinkt
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Expert Answer

Liyana Mansell
Answered 2021-03-06 Author has 97 answers
Step 1
Recall the following.
L{tkf(t)}=(1)kdkdsk(L{f(t)})
L{sin(at)}=as2+a2
Step 2
(a) Obtain the Laplace transform as follows.
(a)f(t)=t2sin(kt)
L{sin(kt)}=ks2+k2
L{t2sin(kt)}=(1)2d2ds2(L{sin(kt)})
=d2ds2(ks2+k2)
=dds(dds(ks2+k2))
=dds(2ks(s2+k2)2)
=2kdds(s(s2+k2)2)
=2k(s2+k2)2dds(s)sdds[(s2+k2)2]((s2+k2)2)2
=2k(s2+k2)24s2[(s2+k2)]((s2+k2)2)2
=2k(s2+k2)4s2(s2+k2)3
=2k(3s2k2)(s2+k2)3
Step 3
(b) Obtain the Laplace transform as follows.
(b)f(t)=tsin(kt)
Lsin(kt)=ks2+k2
L{tsin(kt)}=(1)1d1ds1(L{sin(kt)})
=dds(ks2+k2)
=kdds(1s2+k2)
=k(1(s2+k2)2)dds(s2+k2)

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