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

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

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

This property can be easily extended to more than two functions as shown from the above proof. With the linearity property, Laplace transform can also be called the linear operator.

See examples below.