How Do You Solve L{u(t−2)e3t} Using the Second Shifting Theorem?

[qwerty123z]

looking for the steps to solving this L{u( t − 2)e^{3 t}}. The problem itself looks like a second shifting theorem problem but i don't know how it's done.

answer was given as:

((e⁶)(e^-2s))/(s-3)

 

[yjc]

You can get the answer by applying two properties of the laplace transform. Suppose $x(t) \leftrightarrow X(s)$. Then:
1. $x(t-a) \leftrightarrow X(s)e^{-as}$
2. $x(t)e^{at} \leftrightarrow X(s-a)$

In fact, you should be able to derive these properties in two or three lines, starting from the definition of the laplace transform. The first is derived by redefining $t^{'} = t-a$, the second by redefining $s^{'} = s-a$.