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4. f(t)=e¹+7. 5. f(t)= test. 6. f(t) = tcost. 7. f(t) = t sint. 8. f(t) = et sint.

4. f(t)=e¹+7. 5. f(t)= test. 6. f(t) = tcost. 7. f(t) = t sint. 8. f(t) = et sint.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please only answer problems 6 and 8 of this differential equations.
**Part 1 - Laplace Transform Using the Definition Only** Determine \( \mathcal{L}f(t) \) for the functions \( f(t) \), with \( t > 0 \), defined below. Make sure to specify the frequency domain for \( s \) as well, namely, the domain of convergence of the Laplace transform. ### Graphs: #### 1. Graph 1: - **Description**: A step function. The function \( f(t) \) is at 1 for \( t < 1 \), then jumps to -1 at \( t = 1 \), and remains constant thereafter. #### 2. Graph 2: - **Description**: A ramp function starting at \( t = 1 \). \( f(t) \) begins at 1 and linearly increases to 2 at \( t = 2 \). #### 3. Graph 3: - **Description**: A triangular function. \( f(t) \) starts at 1 and decreases linearly to 0 at \( t = 1 \). ### Functions: 4. \( f(t) = e^{t+7} \). 5. \( f(t) = te^{4t} \). 6. \( f(t) = t \cos t \). 7. \( f(t) = t \sin t \). 8. \( f(t) = e^{-t} \sin t \). Each function listed requires determining its Laplace transform and identifying the region of convergence in the complex plane.
Transcribed Image Text:**Part 1 - Laplace Transform Using the Definition Only** Determine \( \mathcal{L}f(t) \) for the functions \( f(t) \), with \( t > 0 \), defined below. Make sure to specify the frequency domain for \( s \) as well, namely, the domain of convergence of the Laplace transform. ### Graphs: #### 1. Graph 1: - **Description**: A step function. The function \( f(t) \) is at 1 for \( t < 1 \), then jumps to -1 at \( t = 1 \), and remains constant thereafter. #### 2. Graph 2: - **Description**: A ramp function starting at \( t = 1 \). \( f(t) \) begins at 1 and linearly increases to 2 at \( t = 2 \). #### 3. Graph 3: - **Description**: A triangular function. \( f(t) \) starts at 1 and decreases linearly to 0 at \( t = 1 \). ### Functions: 4. \( f(t) = e^{t+7} \). 5. \( f(t) = te^{4t} \). 6. \( f(t) = t \cos t \). 7. \( f(t) = t \sin t \). 8. \( f(t) = e^{-t} \sin t \). Each function listed requires determining its Laplace transform and identifying the region of convergence in the complex plane.
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