How Do You Apply Laplace Transform to Shifted Functions in Calculus?

[jhendren]

Homework Statement

Find Laplace transformation
H(t-1)t^2

Homework Equations

The Attempt at a Solution

[2(e^-s)f(t^2)]/s^3

I'm not sure how to get t^2 in terms of t-1

I know the answer is (2+2s+s^2)/s^3

 

[LCKurtz]

Just express ##t^2## as a finite Taylor series about ##t=1##. Alternatively you can use this formula$$
\mathcal L(u(t-a)f(t)) = e^{-as}\mathcal L(f(t+a)$$which would give, in your case$$
\mathcal L(u(t-1)t^2)=e^{-s}\mathcal L((t+1)^2)
= e^{-s}\mathcal L(t^2+2t+1)$$

 

[jhendren]

can you show me the proof on that formula because it is no where in my book

 

[LCKurtz]

can you show me the proof on that formula because it is no where in my book

$$\mathcal L(u(t-a)f(t)=\int_a^\infty e^{-st}f(t)dt$$Let ##v = t-a##$$
=\int_0^\infty e^{-s(v+a)}f(v+a)dv =e^{-as}\int_0^\infty e^{-sv}f(v+a)dv
=e^{-as}\mathcal L(f(t+a))$$This formula is handy because, as in your problem, the function f(t) isn't usually given in terms of the argument of the step function.