Q2 - tion you obtained in (c); just sketch the graphs to...

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MAE105 Quiz #2 (closed book, closed notes) Name:_________________________ Time: 3:35 to 3:55pm Date: April 24, 2008 Consider the following eigenvalue problem: d 2 φ dx 2 + λ 2 = 0, 0 < x < π , (1) (0) (0) = 0, ( ) = 0 (2) (a) (1 Point) Find the general solution of ODE (1) which involves two arbitrary constants. (b) (1 Point) Apply boundary conditions (2) to obtain two equations for the two constants in the general solution. (c) (1 Point) Using the results in (b), ±nd an equation whose roots are the eigen- values of the problem. (d) (1 Point) Show graphically how one may obtain the eigenvalues of the equa-
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Unformatted text preview: tion you obtained in (c); just sketch the graphs to show you understand the basic appoach. (e) (1 Point) What is the limiting value of n as n becomes very large? For an extra point, start with the ODE (1), multiply through by ( x ), integrate the result from 0 to , use integration by parts and the boundary conditions (2) to obtain an expression for 2 . This is called the Rayleigh quotient....
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