Question 4.2.1: Applying Theorem 4.2.1 Evaluate (a) L^-1 {1/S^5} (b) L^-1{1......
Applying Theorem 4.2.1
Evaluate
(a) \mathscr{L}^{-1}\left\{\frac{1}{s^{5}}\right\} (b) \mathscr{L}^{-1}\left\{\frac{1}{s^{2}+7}\right\}.
(a) To match the form given in part (b) of Theorem 4.2.1, we identify n+15 or n \quad 4 and then multiply and divide by 4 !:
\mathscr{L}^{-1}\left\{\frac{1}{s^{5}}\right\} \quad \frac{1}{4 !} \mathscr{L}^{-1}\left\{\begin{array}{l} {4 !} \\ \hline {s^{5}} \end{array}\right\} \quad \frac{1}{24} t^{4}
(b) To match the form given in part (d) of Theorem 4.2.1, we identify k^{2} \quad 7 and so k \quad \sqrt{7}. We fix up the expression by multiplying and dividing by \sqrt{7} :
\mathscr{L}^{-1}\left\{\frac{1}{s^{2}+7}\right\} \quad \frac{1}{\sqrt{7}} \mathscr{L}^{-1}\left\{\frac{\sqrt{7}}{s^{2}+7}\right\} \quad \frac{1}{\sqrt{7}} \sin \sqrt{7} t.