Question

Find the \(L^{-1} (\frac{s}{(s-1)^7})\).

(a) \(e^{-t} \left (\frac{t^6}{5!}+\frac{t^5}{6!}\right )\)

(b) \(e^t \left (\frac{t^6}{5!}+\frac{t^5}{6!}\right )\)

(c) \(e^t \left (\frac{t^6}{6!}+\frac{t^5}{5!}\right )\)

(d) \(e^{-t} \left (\frac{t^6}{6!}+\frac{t^5}{5!}\right )\)

This question was addressed to me in homework.

This intriguing question originated from General Properties of Inverse Laplace Transform topic in portion Laplace Transform of Engineering Mathematics

Answer 1

The correct choice is (c) \(e^t \left (\frac{t^6}{6!}+\frac{t^5}{5!}\right )\)

To elaborate: In the given question,

=\(L^{-1} \left (\frac{s-1+1}{(s-1)^7}\right)\)

=\(e^t L^{-1} \left (\frac{s+1}{s^7}\right)\)

=\(e^t L^{-1} \left (\frac{1}{s^7}+\frac{1}{s^6}\right)\)

=\(e^t \left (\frac{t^6}{6!}+\frac{t^5}{5!}\right)\)