Laplace

Problem 04 | First Shifting Property of Laplace Transform

Problem 04
Find the Laplace transform of   $f(t) = e^t \sinh 2t$.
 

Solution 04

Problem 03 | First Shifting Property of Laplace Transform

Problem 03
Find the Laplace transform of   $f(t) = e^{-3t} \cos t$.
 

Solution 03

Problem 02 | First Shifting Property of Laplace Transform

Problem 02
Find the Laplace transform of   $f(t) = e^{-5t} \sin 3t$.
 

Solution 02

First Shifting Property | Laplace Transform

First Shifting Property
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   when   $s > a$   then,
 

$\mathcal{L} \left\{ e^{at} \, f(t) \right\} = F(s - a)$

 

In words, the substitution   $s - a$   for   $s$   in the transform corresponds to the multiplication of the original function by   $e^{at}$.
 

Problem 02 | Linearity Property of Laplace Transform

Problem 02
By using the linearity property, show that

$\mathcal{L}(\cosh at) = \dfrac{s}{s^2 - a^2}$

 

Solution 02
$f(t) = \cosh at$

$\displaystyle \mathcal{L}\left\{ f(t) \right\} = \int_0^\infty e^{st} f(t) \, dt$

$\displaystyle \mathcal{L}(\cosh at) = \int_0^\infty e^{st} \cosh at \, dt$
 

But
$\cosh at = \dfrac{e^{at} + e^{-at}}{2}$
 

Thus,
$\displaystyle \mathcal{L}(\cosh at) = \int_0^\infty e^{st} \left( \dfrac{e^{at} + e^{-at}}{2} \right) \, dt$

Problem 01 | Linearity Property of Laplace Transform

Problem 01
Find the Laplace transform of   $f(t) = 5t - 2$.
 

Solution 01

Linearity Property | Laplace Transform

Linearity Property
If   $a$   and   $b$   are constants while   $f(t)$   and   $g(t)$   are functions of   $t$   whose Laplace transform exists, then
 

$\mathcal{L} \left\{ a \, f(t) + b \, g(t) \right\} = a \, \mathcal{L} \left\{ f(t) \right\} + b \, \mathcal{L} \left\{ g(t) \right\}$

 

Proof of Linearity Property
$\displaystyle \mathcal{L} \left\{ a \, f(t) + b \, g(t) \right\} = \int_0^\infty e^{-st}\left[ a \, f(t) + b \, g(t) \right] \, dt$

Table of Laplace Transforms of Elementary Functions

Below are some functions   $f(t)$   and their Laplace transforms   $F(s)$.
 

$f(t)$ $F(s) = \mathcal{L} \left\{f(t)\right\}$
$1$ $\dfrac{1}{s}$
$t$ $\dfrac{1}{s^2}$
$t^2$ $\dfrac{2}{s^3}$
... ...

 

Problem 03 | Laplace Transform by Integration

Problem 03
Find the Laplace transform of   $f(t) = \sin bt$.
 

Problem 03
$\displaystyle \mathcal{L} \left\{f(t)\right\} = \int_0^\infty e^{-st} f(t) \, dt$

$\displaystyle \mathcal{L} (\sin bt) = \int_0^\infty e^{-st} \sin bt \, dt$
 

For   $\displaystyle \int_0^\infty e^{-st} \sin bt \, dt$.

Using integration by parts:   $\displaystyle \int u\,dv = uv - \int v\, du$.   Let

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