Question 5.4.5: Find the Laplace transform of

We first use step functions to write f (t ) with a single formula. Using u(t )−u(t −1) to turn 1 on and off, and similar ideas for t and 2, we have

f(t ) = 1[u(t )−u(t −1)]+t [u(t −1)−u(t −2)]+2u(t −2)
= u(t )+(t −1)u(t −1)+(2−t )u(t −2)

Using the linearity of the Laplace transform, the second shifting property, and familiar transforms,

L[f (t )] = L[u(t )]+L[(t −1)u(t −1)]+L[(2−t )u(t −2)]

= \frac {1}{s}+e^{−s} L[(t +1)−1]+e^{−2s} L[2−(t +2)]

= \frac {1}{s} +e^{−s} L[t]+e^{−2s} L[−t ]

= \frac {1}{s}+ \frac {1}{s^{2}} e^{−s} − \frac {1}{s^{2}} e^{−2s}