Practice Sturm–liouville Problems And Eigenfunctions (18.2)

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6 sample questions

Question 1.
What is a Sturm-Liouville problem?
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Think about the components of a differential equation.
Question 2.
Define eigenvalues in the context of Sturm-Liouville problems.
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Consider their role in eigenfunction solutions.
3Question 3.
What form does a Sturm-Liouville problem take?
- d/dx(p(x)dϕ/dx) + [λw(x) - q(x)]ϕ = 0
- d^2ϕ/dx^2 +λϕ = 0
- ∂u/∂x = f(x)
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Look at how the equations are structured.
4Question 4.
True or False: Eigenfunctions can be non-orthogonal.
- True
- False
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Recall the definition of orthogonality.
Question 5.
Consider a Sturm-Liouville problem with a known weight function w(x) and boundary conditions. How would you derive the eigenfunctions?
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Dive into the boundary conditions to start framing your solutions.
Question 6.
If given a set of eigenvalues λ, how would you check for orthogonality among the corresponding eigenfunctions?
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Remember the definition of the orthogonality condition.

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