How Do You Solve the Unilateral Laplace Transform for Delayed Functions?

[jasonjinct]

Please help me explain how to solve this to find the unilateral laplace transform
x(t)=tu(t) - (t-1)u(t-1) - (t-2)u(t-2) + (t-3)u(t-3)

I know the part tu(t)
as a(t) = u(t) --> 1/s
then b(t) = tu(t) ---> - d/dx a(t) = 1/s^2

But for those (t-1), (t-2) in front u(t), how to solve these?

Please help me, thank you very very much

 

[vela]

Look up how a delay affects the Laplace transform. It's usually one of the properties listed for the transform.

Alternatively, you could derive the relationship for yourself from the definition. Start with

$$L[f(t-a)u(t-a)] = \int_0^\infty f(t-a)u(t-a)e^{-st}\,dt$$

and use the substitution t'=t-a.