(a) With a slight change in notation the transform in (6) is the same as +{ f(t)} s +{ f

Chapter 4, Problem 49

(choose chapter or problem)

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

\(\mathscr{L}\left\{f^{\prime}(t)\right\}=s \mathscr{L}\{f(t)\}-f(0)\),

With \(f(t)=t e^{a t}\), discuss how this result in conjunction with part (c) of Theorem 4.1.1 can be used to evaluate \(\mathscr{L}\left\{t e^{a t}\right\}\).

(b) Proceed as in part (a), but this time discuss how to use (7) with f(t) = t sin kt in conjunction with parts (d) and (e) of Theorem 4.1.1 to evaluate \(\mathscr{L}\{t \sin k t\}\).