Laplace Transforms

[willmoore21]

I have changed this question,


My question now is,

how do I find the laplace transform of tf''(t)?

I know how to find tf(t), and f'(t), and f''(t), but have no idea where to start with tf''(t)? Any hints would be appreciated.

 

[HallsofIvy]

It's pretty direct isn't it? The Laplace transform of \(\displaystyle tf''(t)\) is, by definition, \(\displaystyle \int_0^\infty e^{-st}tf''(t)dt\)
Do an integration by parts, letting \(\displaystyle u= te^{-st}\) and \(\displaystyle dv= f''(t)dt\).

\(\displaystyle du= (e^{-st}- ste^{-st})dt= (1- st)e^{-st}\) and \(\displaystyle v= f'\) so we have
\(\displaystyle (te^{-st}f'(t)|_{t=0}^\infty- \int_0^\infty (1- st)e^{-st})f'(t) dt\)
\(\displaystyle = \int_0^\infty (st- 1)e^{-st})f'(t)dt\)
To integrate that let \(\displaystyle u= (st- 1)e^{-st}\) and [/tex]dv= f'(t)dt[/tex].

Then \(\displaystyle du= (se^{-st}- s(st- 1)e^{-st})dt= (s- s^2t+ s)e^{-st}dt= -s^2te^{-st}dt\) and \(\displaystyle v= f(t)\)
That makes the entire transform
\(\displaystyle (st- 1)e^{-st}f(t)|_{t=0}^\infty+ s^2\int_0^\infty te^{-st}f(t)dt\)
\(\displaystyle = f(0)+ s^2\int_0^\infty te^{-st}f(t)dt\)

That last integral is, of course, the Laplace Transform of tf(t) which you say you know how to do.

 

[willmoore21]

It's pretty direct isn't it? The Laplace transform of \(\displaystyle tf''(t)\) is, by definition, \(\displaystyle \int_0^\infty e^{-st}tf''(t)dt\)
Do an integration by parts, letting \(\displaystyle u= te^{-st}\) and \(\displaystyle dv= f''(t)dt\).

\(\displaystyle du= (e^{-st}- ste^{-st})dt= (1- st)e^{-st}\) and \(\displaystyle v= f'\) so we have
\(\displaystyle (te^{-st}f'(t)|_{t=0}^\infty- \int_0^\infty (1- st)e^{-st})f'(t) dt\)
\(\displaystyle = \int_0^\infty (st- 1)e^{-st})f'(t)dt\)
To integrate that let \(\displaystyle u= (st- 1)e^{-st}\) and [/tex]dv= f'(t)dt[/tex].

Then \(\displaystyle du= (se^{-st}- s(st- 1)e^{-st})dt= (s- s^2t+ s)e^{-st}dt= -s^2te^{-st}dt\) and \(\displaystyle v= f(t)\)
That makes the entire transform
\(\displaystyle (st- 1)e^{-st}f(t)|_{t=0}^\infty+ s^2\int_0^\infty te^{-st}f(t)dt\)
\(\displaystyle = f(0)+ s^2\int_0^\infty te^{-st}f(t)dt\)

That last integral is, of course, the Laplace Transform of tf(t) which you say you know how to do.

Thanks Halls, it was more the integration part to be honest, I didn't know what to put u and v as for parts because I don't think we've covered integration of 3 functions, only triple integration, so I was confused as to what to do. I'll be able to do it now I know what u is. Thanks