Laplace transform | MATHalino reviewer about Laplace transform

Problem 01
Find the Laplace transform of $g(t) = \begin{cases} f(t - 2)^3 & t \gt 2 \\ 0 & t \lt 2 \end{cases}$

Problem 01
Find the Laplace transform of $g(t) = \begin{cases} f(t - 1)^2 & t \gt 1 \\ 0 & t \lt 1 \end{cases}$

Second Shifting Property
If $\mathcal{L} \left\{ f(t) \right\} = F(s)$, and $g(t)
= \begin{cases} f(t - a) & t \gt a \\ 0 & t \lt a \end{cases}$

then,

$\mathcal{L} \left\{ g(t) \right\} = e^{-as} F(s)$

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

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

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

Problem 01
Find the Laplace transform of $f(t) = e^{2t}t^3$.

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