19 - Partial Differential Equations
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Practice Questions
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Define the Laplace Transform.
💡 Hint: What does the transform change about the function?
State the first property of Laplace Transform.
💡 Hint: Think about how we can combine functions.
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Interactive Quizzes
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What does the Laplace Transform do?
💡 Hint: Think about the domain shift it creates.
Laplace Transforms can handle initial conditions naturally. True or False?
💡 Hint: How do we incorporate initial conditions in other methods?
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Challenge Problems
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Given the heat equation \( \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} \) with initial conditions \( u(0, t) = 0 \) and \( u(x, 0) = f(x) \), solve for \( u \) using Laplace transforms.
💡 Hint: Break down the steps into manageable parts.
Consider the wave equation \( \frac{^2u}{^2t} = c^2 rac{^2u}{^2x} \) with conditions \( u(x, 0) = g(x) \) and \( u_t(x, 0) = h(x) \). How would you approach this with Laplace transforms?
💡 Hint: Focus on initial conditions integration into the transform.
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