20.10. Eigenvalues and Eigenfunctions

Try these first

1. 1.

   Define eigenvalue and give an example.

   Hint

   Think about the formula mentioned in class.

2. 2.

   Explain the relationship between eigenvalues and eigenfunctions.

   Hint

   Consider the definitions you learned.

3. 3.

   What is the formula for the eigenvalue α?

   • α = nπ/a
   • α = mπ/b
   • α = n * m

   Hint

   Look for the formula mentioned in class.

4. 4.

   True or False: Eigenfunctions describe the boundary conditions of the system.

   • True
   • False

   Hint

   Think about what each term represents.

5. 5.

   A rectangular membrane has a length of 2m and a width of 1m. Calculate the first three eigenvalues.

   Hint

   Use the formulas for α and β.

6. 6.

   If the initial displacement of a vibrating membrane is given by the function f(x,y) = sin(πx/2)sin(πy), identify the corresponding eigenvalue pairs.

   Hint

   What values of n and m fit the given function?