Problem 03 | Multiplication by Power of t

Problem 03
Find the Laplace transform of   f(t)=t2cos3t.
 

Solution 03
L{tnf(t)}=(1)ndndsnF(s)
 

L(cos3t)=ss2+32

L(cos3t)=ss2+9
 

L(t2cos3t)=(1)2d2ds2[ss2+9]

L(t2cos3t)=dds[(s2+9)(1)s(2s)(s2+9)2]

Problem 02 | Multiplication by Power of t

Problem 02
Find the Laplace transform of   f(t)=tsin3t.
 

Solution 02
L{tnf(t)}=(1)ndndsnF(s)
 

L(sin3t)=3s2+32

L(sin3t)=3s2+9
 

L(tsin3t)=(1)1dds[3s2+9]

L(tsin3t)=[3(2s)(s2+9)2]

L(tsin3t)=6s(s2+9)2

Problem 01 | Multiplication by Power of t

Problem 01
Find the Laplace transform of   f(t)=tcos2t.
 

Solution 01
L{tnf(t)}=(1)ndndsnF(s)
 

L(cos2t)=ss2+22

L(cos2t)=ss2+4
 

L(tcos2t)=(1)1dds[ss2+4]

L(tcos2t)=[(s2+4)(1)s(2s)(s2+4)2]

L(tcos2t)=s2+42s2(s2+4)2