18.3 - General Steps in the Eigenfunction Expansion Method
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Practice Questions
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Define eigenfunction in the context of PDEs.
💡 Hint: Think about functions that remain unchanged apart from scaling.
What step follows after separating variables in a PDE?
💡 Hint: Identify what your variables relate to.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What is the first step in the Eigenfunction Expansion Method?
💡 Hint: Think about how we approach separating different parts.
True or False: Eigenvalues can be complex.
💡 Hint: Recall the properties of eigenvalues in Sturm-Liouville problems.
2 more questions available
Challenge Problems
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Given the boundary conditions for a rod held at two ends and subjected to a heat flux, derive the eigenfunctions and eigenvalues. Next, use them to find the general solution of the heat equation.
💡 Hint: Focus first on applying the correct boundary conditions to obtain eigenvalues.
Consider a wave equation defined on a string of length L with fixed ends. Describe how you would derive the first three eigenfunctions and show how they form a complete basis for the problem.
💡 Hint: Use the knowledge of harmonic frequencies.
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