Laplace Transform (Linear Algebra)

[yesiammanu]

Homework Statement

Evaluate the laplace transform of {t²e^7tsinh(3t)}

Homework Equations

Laplace transform of {tⁿf(t)}=(-1)ⁿdⁿ/ds² * F(s)

The Attempt at a Solution

I've replaced it with (-1)²d²L{e^7tsinh(3t)}

I'm not sure how to proceed, though, as I don't really see how to take the laplace without somehow splitting up these functions. Do I use a unit step function? How would I use it if so? I'm not really sure what to do here

Thanks for any help.

 

[Bipolarity]

The Laplace transform of a function that has been multiplied by an exponential function can be found using the shift theorem for the s-axis. Do you know this theorem?

BiP

 

[yesiammanu]

Is this the theorem you were referring to?

L{e^-asF(s)}=f(t-a)*UnitStep(t-a)

 

[Ray Vickson]

Homework Statement

Evaluate the laplace transform of {t²e^7tsinh(3t)}

Homework Equations

Laplace transform of {tⁿf(t)}=(-1)ⁿdⁿ/ds² * F(s)

The Attempt at a Solution

I've replaced it with (-1)²d²L{e^7tsinh(3t)}

I'm not sure how to proceed, though, as I don't really see how to take the laplace without somehow splitting up these functions. Do I use a unit step function? How would I use it if so? I'm not really sure what to do here

Thanks for any help.

You can write $\sinh(3t) = \frac{1}{2} \left( e^{3t} - e^{-3t}\right),$ and proceed accordingly.