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Section 4 - Laplace Transformations
Section 4.1 - Definition
This section is from Paul's Online Math Notes.
Definition. The Laplace transform of a function is given by the following:
\mathcal{L} \{f(t)\}(s) = F(s) = \int_0^{\infty} e^{-st}f(t) dt
Section 4.2 - Properties
This section is from Paul's Online Math Notes.
The Laplace Transformation is a linear transformation over functions in \mathbb{R}[t]. That is, given a, b \in \mathbb{R}, f(t), g(t) \in \mathbb{R}[t], we know that
\mathcal{L} \{a f(t)\ + b g(t) \}(s) = a F(s) + b G(s)
Section 4.3 - Inverse Laplace Transformation
This section is from Paul's Online Math Notes.
Given F(s), we define the Inverse Laplace Transformation as the following;
f(t) = \mathcal{L}^{-1} \{F(s)\}
Section 4.4 - Step Function
The step/Heaviside function u_c(t) is defined as 0 if t < c, and 1 if t > c.
Alternatively, u(t - c) = H(t - c) is 0 if t < c, and 1 if t > c.
Applying this to the Laplace transform,
\begin{align}
\mathcal{L} \{ u_c(t) f(t-c) \} &= \int_0^{\infty} e^{-st}u_c(t)f(t) dt \\
&= \int_c^{\infty} e^{-st}f(t) dt
\end{align}
If we let u = t - c,
\begin{align}
\mathcal{L} \{ u_c(t) f(t-c) \} &= \int_0^{\infty} e^{-s(u+c)}f(u) du \\
&= \int_0^{\infty} e^{-su}e^{-cs}f(u) du \\
&= e^{-cs} \int_0^{\infty} e^{-su}f(u) du \\
&= e^{-cs} F(s)
\end{align}
Section 4.5 - Laplace Transformation applied to IVPs
This section is from Paul's Online Math Notes.
Theorum. Given a function f(t) with C^n continuity, then
\mathcal{L} \{ f^{(n)} (t) \} = s^n F(s) - s^{n-1} f(0) - s^{n-2} f'(0) - ... - s f^{(n-2)} (0) - f^{(n-1)} (0)
For n=1, 2 we see that
\begin{align}
\mathcal{L} \{ y' \} &= sY(s) - y(0) \\
\mathcal{L} \{ y'' \} &= s^2 Y(s) - s y(0) - y'(0)
\end{align}
We can take the Laplace transformation of an IVP, solve for Y(s), then take the inverse to find the solution.
Section 4.6 - Nonconstant Coefficient IVPs
This section is from Paul's Online Math Notes.