November 15, 2005

This test is open book: Powers ``Boundary Value Problems'' may
be consulted. No other references or notes may be used.

Students may use graphing calculators like TI-83, 85, 86; but
they may NOT use calculators with Computer Algebra Systems,
like TI-89.

Total score = 100.

The heat equation in a laterally insulated bar of length 2 with fixed temperature at one end () and insulated at the other () leads to the eigenvalue problem

- (20 points)
Calculate the eigenvalues and eigenfunctions for this problem.
- (10 points)
Are these eigenfunctions orthogonal, i.e. does

where and are eigenfunctions corresponding to two eigenvalues ? Explain your answer carefully.

- (20 points)
Calculate the eigenvalues and eigenfunctions for this problem.
- The heat equation in a bar of length 2
immersed in a medium of constant
temperature, insulated at its left end ()
and exchanging heat with the medium
along its length and at its right end (), leads (simplifying the constants)
to the eigenvalue problem

- (15 points)
Are the eigenfunctions for this problem orthogonal, i.e. does

where and are eigenfunctions corresponding to two eigenvalues ? Explain your answer carefully. - (20 points)
Calculate the first (the smallest positive) eigenvalue for this problem.

- (15 points)
Are the eigenfunctions for this problem orthogonal, i.e. does
The vibrations of a cord of length 2 fixed at both ends are governed by the wave equation

Suppose that at the cord is not moving, i.e. , and that the graph of the initial position function shows two adjacent equilateral triangles of base 1, like .

- (25 points) Sketch the d'Alembert solution of this problem for .
- (10 points)
The sound spectrum of the resulting vibration will have
a component at frequency
if the eigenfunction corresponding to occurs with
non-zero coefficient in the eigenfunction expansion of .
Which frequencies will be heard in the vibration of the cord with
this initial condition?