**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 \

Name: **Problem 5**: Find \(\mathcal{L}\left\{ e^{7t} \cos^2 t \right\}\)Th
Uploaded: 2024-03-20T06:46:55.000Z
Channel: Bartleby
Description: **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 v