SampleQ_m2 - function 7 Find the Fourier series for the...

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MATH 405: Sample 2nd Midterm Questions 1. Calculate the radius of convergence for the following functions X n =1 n 3 ( n + 1)! x n X m =1 (13 m )! ( m !) 4 x m 2. Evaluate the following Γ(6) / Γ(5) where Γ is the Gamma function. 3. Find a power series solution for the following equation 3 y 00 + (1 + x 3 ) y = 0 4. Laplace’s equation 2 φ = 0 in cylindrical coordinates for φ = φ ( r, θ ) can be written as 2 φ ∂r 2 + 1 r ∂φ ∂r + 1 r 2 2 φ ∂θ 2 = 0 Using separation of variables find the ODE for the radial dependence, and solve it. 5. Show that Bessel’s equation for ν = 0 (which has as solutions J 0 and Y 0 ) is a Sturm- Liouville equation. 6. The function xe x is neither even nor odd. Write it as the sum of an even and an odd
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Unformatted text preview: function. 7. Find the Fourier series for the function f ( x ) = 1-x 2 defined on (-2,2). How would it change if it were defined on (-1 , 3)? 8. Find the eigenfunctions and eigenvalues for the following Sturm-Liouville problem on the interval (0 , 7): y 00 + λy = 0 y (0) = 0 y (7) = 0 What is the weighting function for the orthogonality of the eigenfunctions? 9. Use the Frobenius method if possible to solve x 2 y 00 + 4 xy + ( x 2 + 2) y = 0 If not possible say why not....
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This note was uploaded on 07/23/2008 for the course MATH 405 taught by Professor Belmonte during the Fall '06 term at Penn State.

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