Use the given Laplace transform and the result in to find

Chapter 7, Problem 56E

(choose chapter or problem)

Use the given Laplace transform and the result in Problem 55 to find the indicated Laplace transform. Assume that a and k are positive constants.

(a) \(\mathscr{L}\left\{e^{t}\right\}=\frac{1}{s-1} ; \quad \mathscr{L}\left\{e^{a t}\right\}\)

(b) \(\mathscr{L}\{\sin t\}=\frac{1}{s^{2}+1} ; \quad \mathscr{L}\{\sin k t\}\)

(c) \(\mathscr{L}\{1-\cos t\}=\frac{1}{s\left(s^{2}+1\right)} ; \quad \mathscr{L}\{1-\cos k t\}\)

(d) \(\mathscr{L}\{\sin t \sinh t\}=\frac{2 s}{s^{4}+4} ; \quad \mathscr{L}\{\sin k t \sinh k t\}\)

Text Transcription:

L e^t=1\s-1;L {e^at}

L {sin t}=1/s^2+1;L{sin kt}

L {1-cos t}=1/s(s^2+1);L{1-cos kt}

L {sin t sinh t}=2s/s^4+4;L {sin kt sinh kt}