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# HW03 - APPM 4350/5350 Fourier Series and Boundary Value...

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APPM 4350/5350: Fourier Series and Boundary Value Problems Homework #3 Monday, September 20, 2010 Due: Friday, October 8, 2010 — Note date due! 1. (a) Given the equation for y = y ( x ): x 2 d 2 y dx 2 + (2 b + 1) x dy dx + cy = 0 where b, c are real constants and b 2 > c . (b) Transform the equation from independent variable x to z where x = e z . Hint: the equation should have constant coefficients. (c) Solve for the general solution y = y ( z ) in terms of real functions. (d) Transform the solution back to y = y ( x ) . (e) Show how to get the same solution y = y ( x ) by finding special solutions of the form: y s ( x ) = x p and then finding p . (f) Put the solution in real form if b 2 < c ; find the solution if b 2 = c . 2. In this problem you will solve for the exterior ideal flow around a circle; this problem was discussed in class. However here you will use the velocity potential: u = φ where u is the velocity of the flow. The problem is posed as follows. Find the most general solution φ = φ ( r, θ ) where φ satisfies 2 φ = 0 , ∂φ ∂r ( r = R, θ ) = 0 , φ Ux = Ur cos θ as r

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HW03 - APPM 4350/5350 Fourier Series and Boundary Value...

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