Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Problem 5**: Find \(\mathcal{L}\left\{ e^{7t} \cos^2 t \right\}\)
This problem requires finding the Laplace transform of the given function \( e^{7t} \cos^2 t \). The Laplace transform is an integral transform used to convert a function of a real variable \( t \) (often time) to a function of a complex variable \( s \) (complex frequency).
### Steps to Solve the Problem:
1. **Express the Trigonometric Function with Double Angle Formulas**:
Use the identity for the double angle of cosine:
\[
\cos^2 t = \frac{1 + \cos(2t)}{2}
\]
which simplifies the given function:
\[
e^{7t} \cos^2 t = e^{7t} \cdot \frac{1 + \cos(2t)}{2}
\]
2. **Rewrite the Function**:
Rewrite the function to a more manageable form:
\[
e^{7t} \cos^2 t = \frac{1}{2} e^{7t} + \frac{1}{2} e^{7t} \cos(2t)
\]
3. **Apply Linearity of the Laplace Transform**:
The Laplace transform \(\mathcal{L}\) is linear, so apply it to each term separately:
\[
\mathcal{L}\left\{ \frac{1}{2} e^{7t} + \frac{1}{2} e^{7t} \cos(2t) \right\} = \frac{1}{2} \mathcal{L}\left\{ e^{7t} \right\} + \frac{1}{2} \mathcal{L}\left\{ e^{7t} \cos(2t) \right\}
\]
4. **Find the Laplace Transforms**:
- For \(\mathcal{L}\{ e^{7t} \}\):
\[
\mathcal{L}\left\{ e^{7t} \right\} = \frac{1}{s - 7}, \quad \text{for } s > 7
\]
- For \](https://content.bartleby.com/qna-images/question/24a60763-065d-4deb-b582-ca07220062f2/cbfed147-c394-4765-8bcf-33ca6f7e18be/c1nxw7_processed.jpeg)
Transcribed Image Text:**Problem 5**: Find \(\mathcal{L}\left\{ e^{7t} \cos^2 t \right\}\)
This problem requires finding the Laplace transform of the given function \( e^{7t} \cos^2 t \). The Laplace transform is an integral transform used to convert a function of a real variable \( t \) (often time) to a function of a complex variable \( s \) (complex frequency).
### Steps to Solve the Problem:
1. **Express the Trigonometric Function with Double Angle Formulas**:
Use the identity for the double angle of cosine:
\[
\cos^2 t = \frac{1 + \cos(2t)}{2}
\]
which simplifies the given function:
\[
e^{7t} \cos^2 t = e^{7t} \cdot \frac{1 + \cos(2t)}{2}
\]
2. **Rewrite the Function**:
Rewrite the function to a more manageable form:
\[
e^{7t} \cos^2 t = \frac{1}{2} e^{7t} + \frac{1}{2} e^{7t} \cos(2t)
\]
3. **Apply Linearity of the Laplace Transform**:
The Laplace transform \(\mathcal{L}\) is linear, so apply it to each term separately:
\[
\mathcal{L}\left\{ \frac{1}{2} e^{7t} + \frac{1}{2} e^{7t} \cos(2t) \right\} = \frac{1}{2} \mathcal{L}\left\{ e^{7t} \right\} + \frac{1}{2} \mathcal{L}\left\{ e^{7t} \cos(2t) \right\}
\]
4. **Find the Laplace Transforms**:
- For \(\mathcal{L}\{ e^{7t} \}\):
\[
\mathcal{L}\left\{ e^{7t} \right\} = \frac{1}{s - 7}, \quad \text{for } s > 7
\]
- For \
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