Chapter 7.2, Problem 49E

(a)    With a slight change in notation the transform in (6) is the same as

$L\left\{f^{\prime }\left(t\right)\right\}=sL\left\{f\left(t\right)\right\}-f\left(0\right).$

With $f\left(t\right)=t{e}^{at}$, discuss how this result in conjunction with (c) of Theorem 7.1.1 can be used to evaluate $L\left\{t{e}^{at}\right\}$.

(b)    Proceed as in part (a), but this time discuss how to use (7) with $f\left(t\right)=t\sin kt$ in conjunction with (d) and (e) of Theorem 7.1.1 to evaluate $L\left\{t\sin kt\right\}$.