Subsection 9.2.3 Power Function, \(t^{n}\)
The power function \(t^n\) is another common function type found in differential equations. The Laplace transform of \(t^n\) follows a recursive pattern, which simplifies the computation for higher powers. We’ve already seen that \(\ds\lap{1} = \sfrac{1}{s}\text{.}\) Now, let’s compute the transforms for \(t\) and \(t^2\text{.}\)
If you look in each solution, before computing \(L = 0\text{,}\) you’ll notice a relationship between the Laplace transforms of powers that differ by one. Namely,
\begin{equation*}
\lap{t} = \frac{1}{s}\lap{1} \quad \text{and} \quad \lap{t^2} = \frac{2}{s}\lap{t}\text{,}
\end{equation*}
and if you compute the Laplace transform of \(t^3\text{,}\) you’ll find that
\begin{equation*}
\lap{t^3} = \frac{3}{s}\lap{t^2}\text{.}
\end{equation*}
In general, this recursive pattern continues for any power \(n\) as
\begin{equation*}
\lap{t^n} = \frac{n}{s}\lap{t^{n-1}}\text{.}
\end{equation*}
So if we wanted the Laplace transform of \(t^4\text{,}\) we could find it like so
\begin{align*}
\lap{t^4}
=\amp\ \frac{4}{s}\lap{t^3}\\
=\amp\ \frac{4}{s}\left[\frac{3}{s}\lap{t^2}\right]
= \frac{4\cdot 3}{s^2}\left[\frac{2}{s}\lap{t}\right]
= \frac{4\cdot 3\cdot 2}{s^3}\cdot\frac{1}{s^2}
= \ob{\frac{4\cdot 3\cdot 2\cdot 1}{s^5}}^{\text{factorial}}\text{.}
\end{align*}
This pattern is true for higher powers of \(t\text{,}\) leading to the next laplace transform rule which makes use of the factorial.
Common Laplace Transform (Power).
- \({\LARGE \vphantom{\int}}L_3\)
- \(\ds \lap{ t^n } = \frac{n!}{s^{n+1}}, \quad s >0, \quad n = 1, 2, 3, \ldots \)
Reading Questions Check-Point Questions
1. \(\ds\lap{t^4} = \)\(\ds\frac{\fillinmath{X} !}{s^5}\).
- \(4\)
- Correct! The Laplace transform of \(t^4\) is \(\ds\frac{4!}{s^5}\text{.}\)
- \(5\)
- No, try again.
- \(24\)
- No, notice the factorial in the numerator.
- \(120\)
- No, try again.
2. \(\ds\lap{t^3} = \)\(\ds\frac{6}{\fillinmath{X}}\).
- \(s^4\)
- No, the power of \(s\) in the denominator should be \(4\text{.}\)
- \(s^4\)
- Correct! The Laplace transform of \(t^3\) is \(\ds\frac{6}{s^4}\text{.}\)
- \(s^3\)
- No, the power of \(s\) in the denominator should be \(4\text{.}\)
- \(s^5\)
- No, the power of \(s\) in the denominator should be \(4\text{.}\)
3. \(\ds\lap{\fillinmath{X}} = \ \frac{479001600}{s^{13}}\).
- \(t^9\)
- No, try again.
- \(t^{10}\)
- No, try again.
- \(t^{11}\)
- No, try again.
- \(t^{12}\)
- Correct! The Laplace transform of \(t^{12}\) is \(\ds\frac{479001600}{s^{13}}\text{.}\)
4. \(\ds\lap{t} = \) ?
- \(\ds\frac{1}{s^2}\)
- Correct! The Laplace transform of \(t\) is \(\ds\frac{1}{s^2}\text{.}\)
- \(\ds\frac{1}{s}\)
- No, the power of \(s\) in the denominator is not \(1\text{.}\)
- \(1\)
- No, the Laplace transform of \(t\) is not a constant.
- \(\ds\frac{2}{s^2}\)
- No, there should not be a \(2\) in the numerator.
5. \(\ds\lap{t^2} = \) ?
- \(\ds\frac{1}{s^3}\)
- No, the correct transform is \(\ds\frac{2}{s^3}\text{.}\)
- \(\ds\frac{2}{s^3}\)
- Correct! The Laplace transform of \(t^2\) is \(\ds\frac{2}{s^3}\text{.}\)
- \(\ds\frac{1}{s^2}\)
- No, the correct transform has a \(2\) in the numerator.
- \(\ds\frac{2}{s^2}\)
- No, the power of \(s\) in the denominator should be \(3\text{.}\)
