Ordinary Differential Equations/Laplace Transform

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Definition[edit]

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.

Existence[edit]

Properties[edit]

Linearity[edit]

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[edit]

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}

Laplace Tranform of Higher-Order Derivatives[edit]

Derivatives of the Laplace Transform[edit]

Inverse Laplace Transform[edit]

Definition[edit]

Linearity[edit]