18.2. Sturm–Liouville Problems and Eigenfunctions

Try these first

1. 1.

   What is a Sturm-Liouville problem?

   Hint

   Think about the components of a differential equation.

2. 2.

   Define eigenvalues in the context of Sturm-Liouville problems.

   Hint

   Consider their role in eigenfunction solutions.

3. 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)

   Hint

   Look at how the equations are structured.

4. 4.

   True or False: Eigenfunctions can be non-orthogonal.

   • True
   • False

   Hint

   Recall the definition of orthogonality.

5. 5.

   Consider a Sturm-Liouville problem with a known weight function w(x) and boundary conditions. How would you derive the eigenfunctions?

   Hint

   Dive into the boundary conditions to start framing your solutions.

6. 6.

   If given a set of eigenvalues λ, how would you check for orthogonality among the corresponding eigenfunctions?

   Hint

   Remember the definition of the orthogonality condition.