Question

What is the laplce tranform of the first derivative of a function y(t) with respect to t : y’(t)?

(a) sy(0) – Y(s)

(b) sY(s) – y(0)

(c) s^2 Y(s)-sy(0)-y'(0)

(d) s^2 Y(s)-sy'(0)-y(0)

I had been asked this question in an international level competition.

This key question is from Solution of DE With Constant Coefficients using the Laplace Transform in portion Laplace Transform of Engineering Mathematics

Answer 1

Right option is (b) sY(s) – y(0)

To explain I would say: Let \(f(t) = y(t) \)

\(L[f’(t)] = \int_0^∞ e^{-st} f'(t)dt \)

\( = e^{-st} f(t)(from \, 0 \, to \, \infty) – \int_0^∞ (-s) e^{-st} f(t)dt \)

\( = -f(0) + s\int_0^{\infty} e^{-st} f(t)dt \)

\( = -f(0) + sF(s) \)

\( = sY(0) -y(0) \).