Question

what is the laplace transform of the first derivative of the function y(t) with resptive t : y(t) mcq​

Answer 1

Answer:

what is the laplace transform of the first derivative of the function y(t) with resptive t : y(t) mcq

Answer 2

The Laplace transform of the first derivative of the function y(t) with respective t is sY(s) – y(0) where Y(s) is Laplace transform of y(t)

Given :

The function y(t)

To find :

The Laplace transform of the first derivative of the function y(t) with respective t

Solution :

Step 1 of 2 :

Define Laplace transform

The Laplace Transform of y(t), denoted by L{y(t)} and defined as :

$\displaystyle \sf{ L\{y(t)\} =Y(s) = \displaystyle \sf\int\limits_{0}^{\infty} e^{-st} y(t)\, dt }$

Step 2 of 2 :

Find Laplace transform of the first derivative of the function y(t) with respective t

$\displaystyle \sf{ L\{y'(t)\} }$

$\displaystyle \sf= \int\limits_{0}^{\infty} e^{-st} \: y'(t)\, dt$

$\displaystyle \sf= e^{-st} \: y(t) \bigg|_{0}^{\infty} - ( - s)\int\limits_{0}^{\infty} e^{-st} \: y(t)\, dt$

$\displaystyle \sf=0 - y(0) + s\int\limits_{0}^{\infty} e^{-st} \: y(t)\, dt$

$\displaystyle \sf= - y(0) + sY(s)$

$\displaystyle \sf= sY(s) - y(0)$