Select the missing piece from each Laplace transform.
Section The Laplace Transform Reference Guide
In this final section, we summarize everything weβve done. Below, you will find tables for the most common Laplace transforms and the properties used to combine and manipulate them. These will provide a useful resource in the next chapter where we begin solving differential equations using the Laplace transform method.
Subsection Laplace Transform Tables
We have done a lot of work computing common transforms and deriving the properties of the Laplace transform. Now, we can summarize all of this information in tables for quick reference.
We begin with the specific transforms of exponentials, polynomial terms, sine/cosine, and the products of these functions. In later chapters, we will encounter new types of functions that we need to transform, so this table will continue to grow as we go.
| Function (\(t\)-Domain)
\begin{equation*}
f(t)
\end{equation*}
|
Laplace Transform (\(s\)-Domain)
\begin{equation*}
\lap{f(t)} = F(s)
\end{equation*}
|
Existence Condition |
|
| L\(_1\) |
\begin{equation*}
1
\end{equation*}
|
\begin{equation*}
\frac{1}{s}
\end{equation*}
|
\begin{equation*}
s > 0
\end{equation*}
|
| L\(_2\) |
\begin{equation*}
e^{at}
\end{equation*}
|
\begin{equation*}
\frac{1}{s - a}
\end{equation*}
|
\begin{equation*}
s > a
\end{equation*}
|
| L\(_3\) |
\begin{equation*}
t^n
\end{equation*}
|
\begin{equation*}
\frac{n!}{s^{n+1}}
\end{equation*}
|
\begin{equation*}
s > 0
\end{equation*}
|
| L\(_4\) |
\begin{equation*}
\sin(bt)
\end{equation*}
|
\begin{equation*}
\frac{b}{s^2+b^2}
\end{equation*}
|
\begin{equation*}
s > 0
\end{equation*}
|
| L\(_5\) |
\begin{equation*}
\cos(bt)
\end{equation*}
|
\begin{equation*}
\frac{s}{s^2+b^2}
\end{equation*}
|
\begin{equation*}
s > 0
\end{equation*}
|
| L\(_6\) |
\begin{equation*}
e^{at}\ t^n
\end{equation*}
|
\begin{equation*}
\frac{n!}{(s-a)^{n+1}}
\end{equation*}
|
\begin{equation*}
s > a
\end{equation*}
|
| L\(_7\) |
\begin{equation*}
e^{at}\sin(bt)
\end{equation*}
|
\begin{equation*}
\frac{b}{(s-a)^2+b^2}
\end{equation*}
|
\begin{equation*}
s > a
\end{equation*}
|
| L\(_8\) |
\begin{equation*}
e^{at}\cos(bt)
\end{equation*}
|
\begin{equation*}
\frac{s-a}{(s-a)^2+b^2}
\end{equation*}
|
\begin{equation*}
s > a
\end{equation*}
|
Checkpoint 200. πβ Whatβs Missing in each Transform?
The next set of transforms are more general rules you can apply in different situations.
| Function (\(t\)-Domain) | Laplace Transform (\(s\)-Domain) | Existence | |
\begin{equation*}
f(t)
\end{equation*}
|
\begin{equation*}
\lap{f(t)} = F(s)
\end{equation*}
|
Condition | |
| R\(_{1}\) |
\begin{equation*}
f'(t)
\end{equation*}
|
\begin{equation*}
sF(s) - f(0)
\end{equation*}
|
\begin{equation*}
s > 0
\end{equation*}
|
| R\(_{2}\) |
\begin{equation*}
f''(t)
\end{equation*}
|
\begin{equation*}
s^2F(s) - sf(0) - f'(0)
\end{equation*}
|
\begin{equation*}
s > 0
\end{equation*}
|
| R\(_{3}\) |
\begin{equation*}
f'''(t)
\end{equation*}
|
\begin{equation*}
s^3F(s) - s^2f(0) - sf'(0) - f''(0)
\end{equation*}
|
\begin{equation*}
s > 0
\end{equation*}
|
| R\(_{4}\) |
\begin{equation*}
e^{at} f(t)
\end{equation*}
|
\begin{equation*}
F(s-a)
\end{equation*}
|
\begin{equation*}
s > 0
\end{equation*}
|
| R\(_{5}\) |
\begin{equation*}
t^n f(t)
\end{equation*}
|
\begin{equation*}
(-1)^n \frac{d^n}{ds^n}\Big[F(s)\Big]
\end{equation*}
|
\begin{equation*}
s > 0
\end{equation*}
|
Finally, we list the Laplace transform properties. It is short for now, but we will add more to this table later.
Subsection Next Step: Solving Differential Equations
Everything youβve learned in this chapter prepares you to solve differential equations using the Laplace transform. In the next chapter, you will see how these tools turn the process of solving differential equations into solving simple algebraic equations, especially when initial conditions are involved.
Subsection π€ Wrap-Up
Check Your Understanding.
Checkpoint 203. π€π Separation of Variables Reading Questions.
(a) π€π Transform of \(\sin(5t)\).
\(\ds\lap{\sin(5t)} = \) ?
- \(\ds\frac{5}{s^2 + 25}\)
- Correct! This follows the formula \(\lap{\sin(bt)} = \frac{b}{s^2 + b^2}\) with \(b = 5\text{.}\)
- \(\ds\frac{s}{s^2 + 25}\)
- Incorrect. Thatβs the transform of \(\cos(5t)\text{,}\) not \(\sin(5t)\text{.}\)
- \(\ds\frac{1}{s^2 + 25}\)
- Incorrect. The frequency coefficient \(5\) must appear in the numerator.
- \(\ds\frac{5}{s^2 + 5^2}\)
- This is mathematically correct but not simplified. Choose the version with \(25\) in the denominator.
(b) π€π Transform of \(\cos(-3t)\).
\(\ds\lap{\cos(-3t)} = \) ?
- \(\ds\frac{3}{s^2 + 9}\)
- Incorrect. The numerator for cosine should be \(s\text{,}\) not the frequency.
- \(\ds\frac{s}{s^2 - 9}\)
- Incorrect. The denominator should be \(s^2 + 9\text{,}\) not \(s^2 - 9\text{.}\)
- \(\ds\frac{s}{s^2 + 9}\)
- Correct! Cosine is even, so \(\cos(-3t)\) = \(\cos(3t)\text{,}\) and its transform is \(\ds\frac{s}{s^2 + 9}\text{.}\)
- \(\ds\frac{3}{s^2 - 9}\)
- Incorrect. Both the numerator and denominator are wrong for cosine transforms.
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