mt1 - b) Put this equation in self-adjoint form. c)...

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PHYS 245 Mathenmatical Methods of Physics II Fall 2008 Midterm I Instructions Please read the questions carefully before writing your solutions. Use separate sheets for di±erent questions. Make sure your handwriting is legible. Make sure that your line of reasoning for a given problem is clearly re²ected in your solution. You may loose points if the above guidelines are not satisfactorily followed. Let me remind you that cheating is a capital o±ense. Disciplinary action will be taken if the con³dence entrusted in you is breached.
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1) Transform the second order ODE y ′′ + P ( x ) y + Q ( x ) y = 0 by the substitution y = ze i x P ( t ) dt to Fnd the resulting di±erential equation for z . 2) Consider the following ODE (1 - x 2 ) U ′′ n ( x ) - 3 xU n + n ( n + 2) U n ( x ) = 0 a) Locate the singular points and show whether they are regular or irregular.
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Unformatted text preview: b) Put this equation in self-adjoint form. c) Identify the eigenvalue. d) Identify the weight function. 3) One solution of Laguerre’s di±erential equation xy ′′ + (1-x ) y ′ + ny = 0 for n = 0 is y 1 ( x ) = 1. Develop a second, linearly independent solution. 4) Construct the Green’s function for the operator d 2 /dx 2 and the boundary conditions y (0) = 0 and y ′ (1) = 0. Consider now the equation y ′′ ) x ) + λy ( x ) = 0 with the same boundary conditions. Obtain the solutions. Verify that the solutions satisfy y ( x ) = λ I 1 G ( x, t ) y ( t ) dt Good Luck...
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mt1 - b) Put this equation in self-adjoint form. c)...

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