Properties of Laplace Transform

Constant Multiple
If   $a$   is a constant and   $f(t)$   is a function of   $t$,   then
 

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

 

Example: $\mathcal{L} (4 \cos t) = 4 \, \mathcal{L} (\cos t)$
 

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

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

 

Example: $\mathcal{L} (3t^2 - 4t + 9) = 3 \, \mathcal{L} (t^2) - 4 \, \mathcal{L} (t) + 9 \, \mathcal{L} (1)$
 

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

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

 

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,

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

 

Change of Scale Property
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   then,
 

$\mathcal{L} \left\{ f(at) \right\} = \dfrac{1}{a} F \left( \dfrac{s}{a} \right)$

 

Multiplication by Power of $t$
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   then,
 

$\mathcal{L} \left\{ t^n f(t) \right\} = (-1)^n \dfrac{d^n}{ds^n} F(s) = (-1)^n F^{(n)}(s)$

where   $n = 1, \, 2, \, 3, \, ...$
 

Division by $t$
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   then,
 

$\displaystyle \mathcal{L} \left\{ \dfrac{f(t)}{t} \right\} = \int_s^\infty F(u) \, du$

provided   $\displaystyle \lim_{t \rightarrow 0} \left[ \dfrac{f(t)}{t} \right]$   exists.
 

Transforms of Derivatives
The Laplace transform of the derivative   $f'(t)$   exists when   $s > a$,   and
 

$\mathcal{L} \left\{ f'(t) \right\} = s\mathcal{L} \left\{ f(t) \right\} - f(0)$

 

In general, the Laplace transform of nth derivative is

$\mathcal{L} \left\{ f^n(t) \right\} = s^n\mathcal{L} \left\{ f(t) \right\} - s^{n - 1}f(0) - s^{n - 2}f'(0) - s^{n - 3}f''(0) - \, ... \, - f^{n - 1}(0)$