Subsection 9.3.2 Multiplication by \(e^{at}\)
The translation property, also known as the first shifting theorem, allows us to handle functions multiplied by an exponential term, \(e^{at}\text{.}\) This property is particularly useful for simplifying the Laplace transforms of products of exponential functions and other functions, such as sine, cosine, or polynomials.
Example 9.
\(\ \ \)\(\lap{e^{7t}\cos(3t)} \text{.}\)Solution.
By the definition of the Laplace transform, we have:
\begin{align*}
\lap{e^{7t}\cos(3t)}
=\amp \int_0^{\infty} e^{-st} e^{7t}\cos(3t)\ dt\\
=\amp \int_0^{\infty} e^{-st+7t} \cos(3t)\ dt\\
=\amp \int_0^{\infty} e^{-(s-7)t} \cos(3t)\ dt\\
=\amp \ub{\int_0^{\infty} e^{-(s_0)t} \cos(3t)\ dt}_{\lap{\cos(3t)}}, \quad \text{where } s_0 = s-7\\
=\amp\ \frac{s_0}{s_0^2 + 9} \qquad (s_0 \gt 0)\\
=\amp\ \frac{s-7}{(s-7)^2 + 9} \qquad (s-7 \gt 0 \text{ or } s \gt 7)
\end{align*}
Thus, the Laplace transform of \(e^{7t}\cos(3t)\) is:
\begin{equation*}
\ds \lap{e^{7t}\cos(3t)} = \frac{s-7}{(s-7)^2 + 9}, \quad s \gt 7.
\end{equation*}
The translation property can be generalized for any function \(f(t)\) multiplied by \(e^{at}\text{.}\) The property is formally stated as:
\begin{equation*}
\lap{ e^{at} f(t) } = F(s-a),
\end{equation*}
where \(F(s)\) is the Laplace transform of \(f(t)\text{.}\)
Laplace Transform Property \(P_2\).
Let \(F(s) = \lap{f(t)}\text{.}\)
- \(P_2\)
- \(\ds \lap{ e^{at} f(t) } = F(s-a), \quad a \) is a constant.
By applying this property to the functions \(t^n\text{,}\) \(\cos(bt)\text{,}\) and \(\sin(bt)\text{,}\) we can derive additional common Laplace transforms:
Common Laplace Transforms \(L_6-L_8\).
- \({\LARGE \vphantom{\int}}L_6\)
- \(\lap{ t^n e^{at} } = \ds\frac{n!}{(s-a)^{n+1}}, \quad s >a \)
- \({\LARGE \vphantom{\int}}L_7\)
- \(\ds \lap{ e^{at}\sin(bt) } = \frac{b}{(s-a)^2+b^2}, \quad s >a \)
- \({\LARGE \vphantom{\int}}L_8\)
- \(\ds \lap{ e^{at}\cos(bt) } = \frac{s-a}{(s-a)^2+b^2}, \quad s >a\)
Reading Questions Check-Point Questions
1. \(\quad \lap{e^{5t} \sin(2t)} = \) .
- \(\ds\frac{2}{(s-5)^2+4}\)
Correct! The translation property shifts the transform of \(\sin(2t)\) by 5 units, giving \(2/(s-5)^2+4\)
- \(\ds\frac{5}{(s-2)^2+4}\)
No, the translation property involves shifting the transform of \(\sin(2t)\) by 5, not by 2.
- \(\ds\frac{2}{(s+5)^2+4}\)
No, the correct shift should be \(s-5\text{,}\) not \(s+5\text{.}\)
- \(\ds\frac{5}{(s-2)^2+2}\)
No, the denominator should have \(2^4 b^2=4\text{,}\) not 2.
2. \(\quad \lap{e^{2t} t^3} = \) .
- \(6/(s-2)^4\)
Correct! The translation property applied to \(t^3\) gives \(6/(s-2)^4\)
- \(6/(s+2)^4\)
No, the correct shift should be \(s-2\text{,}\) not \(s+2\text{.}\)
- \(3/(s-2)^3\)
No, the denominator should have \(3^4=6\text{,}\) not 3.
- \(3/(s+2)^3\)
No, the correct shift should be \(s-2\) and the power should be 4.
3. \(\quad \lap{e^{4t} \cos(5t)} = \) .
- \(4/(s-4)^2+25\)
Correct! The Laplace transform of \(4 \cos 5 e^{4t} \cos(5t)\) is \(4/(s-4)^2+25\)
- \(4/(s+4)^2+25\)
No, the correct shift should be \(s-4\text{,}\) not \(s+4\text{.}\)
- \(5/(s-5)^2+16\)
No, the shift should be \(s-4\) and the denominator should have \(25=5^2\)
- \(4/(s-4)^2+16\)
No, the correct denominator should be \(4^2 25/(s-4)^2+25\text{,}\) not \(16^2\)
4. The translation property only works for exponential functions multiplied by sine and cosine functions.
True.
False. The translation property applies to any function \(f(t)\) multiplied by an exponential term \(e^{at}\)
False.
False. The translation property applies to any function \(f(t)\) multiplied by an exponential term \(e^{at}\)
5. If \(\ds\lap{f(t)}=\frac{1}{s(s+1)}\text{,}\) what is the Laplace transform of \(e^{2t}f(t)\text{?}\)
- \(\ds\frac{1}{(s+2)(s+1)}\)
No, the correct shift should be \(s-2\text{,}\) not \(s+2\text{.}\)
- \(\ds\frac{1}{s(s+2)}\)
No, the correct shift should be \(s-2\text{,}\) not \(s\text{.}\)
- \(\ds\frac{1}{(s-2)(s+2)}\)
No, the correct shift should be \(s-2\text{,}\) not \(s+2\text{.}\)
- \(\ds\frac{1}{(s-2)(s+1)}\)
Correct! The translation property shifts the transform of \(f(t)\) by 2 units, giving \(1/(s-2)(s+1)\)
