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# What is the formula for c n or simpler what is a

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, what is the formula for c n ? Or simpler: What is a regular Sturm-Liouville problem? (c) Fillers. Short questions that take about a minute or less to answer correctly, if you know the answer. Example: If x 3 = n =1 b n sin nπx 2 for 0 < x < 2; what does n =1 b n sin 7 2 equal? The material of the course can be divided into the following topics, which correspond to Chapters 2, 3, 5, 6, 7, 8 of the textbook: I. Fourier series. Things to know: What is a Fourier series, how does one compute the coeﬃcients, convergence facts. Typical problem: A relatively simple function is given, say f ( x ) = x 2 in the interval [ - 1 , 2], or a functions with a few more jumps in some other interval, and you may be asked to expand it into a Fourier series, asked questions about its convergence. Similarly for sine and cosine series. II. Sturm-Liouville Problems. Know the difference between regular, singular and periodic. Main properties of eigenvalues and eigenfunctions. Rayleigh quotient. Here are some problems. Exercises 4, 5 are good practice exercises, but one could argue that it might not be fair to have similar ones in the final. One could argue. 1. Consider the problem y ′′ - xy + λ ( x 2 + 1) y = 0 , 0 < x < 1 , y (0) = 0 , y (1) = 0 . Show that it is a regular Sturm-Liouville problem. Be sure to state clearly what p ( x ) , q ( x ), and σ ( x ) are.

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2 2. Consider the Sturm-Liouville problem y ′′ + 4 y + 5 y + λy = 0 , 0 x 2 π, y (0) = 0 , y (2 π ) = 0 . (a) Find all eigenvalues and corresponding eigenfunctions. (b) Suppose the function f ( x ) = { x, 0 < x < π, 0 , π x < 2 π, is expanded into a series of eigenfunctions. Write out a formula for the coeﬃcients. Do not compute any integrals, except if you have time to spare. 3. Compute all eigenfunctions and eigenvalues of the following regular Sturm-Liouville problem. y ′′ + λy = 0 , 0 < x < 1 , y (0) - y (0) = 0 , y (1) = 0 . 4. Compute all eigenfunctions and eigenvalues of the following regular Sturm-Liouville problem.
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