I'm not sure where I need to start here. I know the laplace of the heaviside is (e^-as)/s, but I dont know what needs to be done with the t^3 since it is multiplied by the heaviside.
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1st : you need to write t^3 in terms of ( t - 4 )...
t^3 = 64 + 48 [ t - 4 ] + 24 [ t-4]² / 2 + 6 [ t - 4 ]^3 / 6 ---> F(t) = 64 + 48 t + 24 t² +t^3
L{ H(t-4)F(t) } = e^(-4s) { 64 / s + 48 / s² + 12 / s^3 + 1 / s^4 }
t^3 = 64 + 48 [ t - 4 ] + 24 [ t-4]² / 2 + 6 [ t - 4 ]^3 / 6 ---> F(t) = 64 + 48 t + 24 t² +t^3
L{ H(t-4)F(t) } = e^(-4s) { 64 / s + 48 / s² + 12 / s^3 + 1 / s^4 }
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Not hard to do from the definition of the Laplace transform;
if you care about that, look at Paul's Online Math Notes.
However, the answer is that
H(t-c) f(t) transforms to e^(-cs) F(s).
Since the Laplace transform of t^3 is 6/s^4,
the transform of the function you wrote is 6e^(-4s)/s^4.
if you care about that, look at Paul's Online Math Notes.
However, the answer is that
H(t-c) f(t) transforms to e^(-cs) F(s).
Since the Laplace transform of t^3 is 6/s^4,
the transform of the function you wrote is 6e^(-4s)/s^4.