Laplace Transformations are an integral transformation used to transform differential equations into algebraic equations. We can use them to solve non-homogeneous differential equations of higher order. In general, integral transformations are in the form:
\( \displaystyle \int_\alpha^{\beta} {K(s,t) f(t)dt} \)
Note that the independent variable is originally \(t\) and is then \(s\) after the transformation. The function \(K(s,t)\) is called a kernal. For the Laplace Transform:
\( K(s,t) = e^{-st}\)
The notation for the Laplace Transformation is:
\( \mathcal{L} \{f(t)\} = F(s) = \int_0^{\infty} {e^{-st} f(t)dt}, \; t \ge 0\)
The function being transformed is typically written in lower case and the transformed function is in upper case. The original function can be referred to as the inverse Laplace Transformation denoted as:
\( \mathcal{L}^{-1} \{ Y(s) \} = y(t) \)
The Laplace Transform is a linear operator, thus:
\( \mathcal{L}\{ a f(t) + bg(t) \} = aF(s) + bG(s)\)
Although the definition can be used to derive the Laplace Transformation of common functions, the table below can be used.
| Function | Laplace |
| \( 1 \) | \( \frac{1}{s} \) |
| \( t^n, n \epsilon \mathbb R \) | \( \frac{n!}{s^{n+1}} \) |
| \( e^{at} \) | \( \frac{1}{s-a} \) |
| \( \sin{(kt)} \) | \( \frac{k}{s^2 + k^2} \) |
| \( \cos{(kt)} \) | \( \frac{s}{s^2 + k^2} \) |
| \( \sinh{(kt)} \) | \( \frac{k}{s^2 - k^2} \) |
| \( \cosh{(kt)} \) | \( \frac{s}{s^2 - k^2} \) |
Determine \( \mathcal{L} \{ e^{3t} +\frac{1}{2}t - 4 \}. \)
Determine \( \mathcal{L}^{-1} \{ \cfrac{3s-1}{s^2+16} \}. \)
Our goal is to use Laplace Transformations to solve differential equations. Using the definition of a Laplace Transformation, we can show that the Laplace Transformation becomes an algebraic equation in terms of initial values:
\( \mathcal{L} \{ f^{(n)}(t) \} = s^nF(s) - s^{n-1} f(0) - s^{n-2} f^{'}(0) - \dots - f^{(n-1)}(0) \)
Determine \( \mathcal{L}^{-1} \{ 3y^{''} - 2y \} \) given \(y^{'}(0) = 2, y^{''}(0) = 0 \).