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k x (t) a b t

k x (t) a b t

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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Determine the laplace transform of the figure shown below with a, b, and k as constraints.

The diagram is a graph representing a piecewise linear function \( x(t) \) with respect to \( t \). - The horizontal axis is labeled as \( t \). - The vertical axis is labeled as \( x(t) \). Key Features: 1. **Graph Shape**: The graph is a symmetrical triangle. - It starts from the origin point (0,0). - It rises linearly to a peak point at \( (a, k) \). - It then decreases linearly back to the horizontal axis at point \( (b, 0) \). 2. **Points**: - The peak of the triangle is at time \( t = a \) and value \( x(t) = k \). - The base of the triangle runs from \( t = 0 \) to \( t = b \). 3. **Dashed Lines**: - A horizontal dashed line extends from the peak at \( (a, k) \) to the vertical axis, indicating the maximum value \( k \). - A vertical dashed line drops from the peak at \( (a, k) \) to the horizontal axis, indicating the time \( t = a \). This type of graph is often used to represent a time-dependent process that grows to a certain level and then declines symmetrically, similar to a triangle wave.
Transcribed Image Text:The diagram is a graph representing a piecewise linear function \( x(t) \) with respect to \( t \). - The horizontal axis is labeled as \( t \). - The vertical axis is labeled as \( x(t) \). Key Features: 1. **Graph Shape**: The graph is a symmetrical triangle. - It starts from the origin point (0,0). - It rises linearly to a peak point at \( (a, k) \). - It then decreases linearly back to the horizontal axis at point \( (b, 0) \). 2. **Points**: - The peak of the triangle is at time \( t = a \) and value \( x(t) = k \). - The base of the triangle runs from \( t = 0 \) to \( t = b \). 3. **Dashed Lines**: - A horizontal dashed line extends from the peak at \( (a, k) \) to the vertical axis, indicating the maximum value \( k \). - A vertical dashed line drops from the peak at \( (a, k) \) to the horizontal axis, indicating the time \( t = a \). This type of graph is often used to represent a time-dependent process that grows to a certain level and then declines symmetrically, similar to a triangle wave.
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