8.1.5.1 - Example 2
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Practice Questions
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What is the division by t rule in Laplace transforms?
💡 Hint: Think about what integration means in the s-domain.
Name one application of the division by t property.
💡 Hint: Consider fields where time-varying inputs are encountered.
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Interactive Quizzes
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What does the division by t rule correspond to in the s-domain?
💡 Hint: Link division operations with their respective integration implications.
The division by t rule is applicable only when the function is:
💡 Hint: Consider the continuity conditions for applying the transformations.
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Challenge Problems
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Using the division by t rule, find the Laplace transform of a piecewise function defined by \( f(t) = \sin(at), t \leq 1 \) and \( 1, t > 1 \).
💡 Hint: Break the function into manageable parts before applying the rule.
Demonstrate how the division by t rule can aid in resolving the differential equation \( y'' + 2y' + y = \frac{1}{t} \) with initial conditions.
💡 Hint: Analyze the terms separately and employ Laplace properties effectively.
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