# Ordinary Differential Equations/Laplace Transform

## Definition

Let $f(t)$ be a function on $[0,\infty)$. The Laplace transform of $f$ is defined by the integral

$F(s) = \mathcal{L}\{f\}(s) = \int_0^{\infty} e^{-st} f(t) dt\,.$

The domain of $F(s)$ is all values of $s$ such that the integral exists.

## Properties

### Linearity

Let $f$ and $g$ be functions whose Laplace transforms exist for $s > \alpha$ and let $a$ and $b$ be constants. Then, for $s > \alpha$,

$\mathcal{L}\{af + bg\} = a \mathcal{L}\{f\} + b \mathcal{L}\{g\}\,,$

which can be proved using the properties of improper integrals.

### Shifting in s

If the Laplace transform $\mathcal{L}\{f\}(s) = F(s)$ exists for $s > \alpha$, then

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

for $s > \alpha + a$.

Proof.

\begin{align} \mathcal{L}\{e^{at} f(t)\}(s) &{} = \int_0^{\infty} e^{-st} e^{at} f(t) dt \\ &{} = \int_0^{\infty} e^{-(s-a)t} f(t) dt \\ &{} = F(s - a)\,. \end{align}