22. Starting with the transform use formula (6) for the

Chapter 7, Problem 22E

(choose chapter or problem)

Starting with the transform \(\mathscr{L}\{1\}(S)=1 / s\) use formula (6) for the derivatives of the Laplace transform to show that \(\mathscr{L}\{t\}(S)=1 / s^{2}, \mathscr{L}\left\{t^{2}\right\}(s)=2 ! / s^{3}\), and, by using induction, that \(\mathscr{L}\left\{t^{n}\right\}(S)=n ! / s^{n+1}, n=1,2, \ldots\)

Equation transcription:

Text transcription:

{L}{1}(S)=1 / s

{L}{t}(S)=1 / s^{2},{L}{t^{2}}(s)=2 ! / s^{3}

r{L}{t{n}}(S)=n ! / s^{n+1}, n=1,2, ldots