Question

Show that if \begin{equation} F(s) = L\{f(x)\} = \int_0^\infty e^{-sx} f(x)dx \end{equation} then \begin{equation} (i) \quad F'(s) = -L\{xf(x)\} \end{equation} \begin{equation} (ii) \quad F''(s) = L\{x^2 f(x)\} \end{equation}

          Show that if \begin{equation*} F(s) = L\{f(x)\} = \int_0^\infty e^{-sx} f(x)dx \end{equation*} then \begin{equation*} (i) \quad F'(s) = -L\{xf(x)\} \end{equation*} \begin{equation*} (ii) \quad F''(s) = L\{x^2 f(x)\} \end{equation*}

$Show that if F(s) = L{f(x)} = ∫0^∞ e^-sx f(x)dx then (i) F'(s) = -L{xf(x)} (ii) F”(s) = L{x^2 f(x)}$

Added by Mary G.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Aman Gupta

10/14/2023

Step 1

Step 1: To find F'(s), we differentiate F(s) with respect to s using the Leibniz integral rule: F'(s) = d/ds [∫_0^∞ e^(-sx)f(x)dx] F'(s) = ∫_0^∞ d/ds [e^(-sx)f(x)]dx F'(s) = ∫_0^∞ -xe^(-sx)f(x)dx Show more…

Show all steps