Definition of Laplace Transform

Differential Equations · Learn by Concept

Help Questions

Differential Equations › Definition of Laplace Transform

1 - 1

Find the Laplace Transform for the following function.

$\mathcal{L}\begin{Bmatrix} e^{-5t} \end{Bmatrix}$

$\mathcal{L}\begin{Bmatrix} e^{-5t} \end{Bmatrix}=\frac{1}{s+5}, s>-5$

CORRECT

$\mathcal{L}\begin{Bmatrix} e^{-5t} \end{Bmatrix}=\frac{1}{s-5}, s>-5$

$\mathcal{L}\begin{Bmatrix} e^{-5t} \end{Bmatrix}=\frac{1}{s+3}, s>-5$

$\mathcal{L}\begin{Bmatrix} e^{-5t} \end{Bmatrix}=\frac{1}{s+5}, s>5$

$\mathcal{L}\begin{Bmatrix} e^{-5t} \end{Bmatrix}=\frac{1}{s+5}, s<-5$

To find the Laplace Transform for the following function

$\mathcal{L}\begin{Bmatrix} e^{-5t}\end{Bmatrix}$

use the transforms for basic functions which state,

$\mathcal{L}\begin{Bmatrix} 1 \end{Bmatrix}=\frac{1}{s}$

$\mathcal{L}\begin{Bmatrix} e^{at} \end{Bmatrix}=\frac{1}{s-a}$

$\mathcal{L}\begin{Bmatrix} e^{-5t} \end{Bmatrix}=\int_0^\infty e^{-st}e^{-5t}dt=\int_0^\infty e^{-(s+5)t} dt$

$\=\frac{-e^{-(s+5)t}}{s+5}\bigg|_0^\infty \\=\frac{1}{s+5}, s>-5$