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test1-a

# C using the ourier series itself prove that s k 0 1 2

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c) Using the ±ourier series itself, prove that s k =0 1 (2 k + 1) 2 = π 2 8 8) (10 points) a) Let a = b = 1, f 1 ( x ) = 0, f 2 ( x ) = x , g 1 ( y ) = 0, and g 2 ( y ) = y . Solve the Laplace equation with boundary conditions u ( x, 0) = f 1 ( x ), u ( x, 1) = f 2 ( x ), u (0 , y ) = g 1 ( y ), and u (1 , y ) = g 2 ( y ). b) Of course, your answer in part a) is really, really ugly. But there is a nice, pretty answer for u ( x, y ). Can you Fnd a really simple expression of u ( x, y )? 9) (10 points) Consider the PDE 2 u ∂x 2 + 2 u ∂x∂y + 2 u ∂y 2 = 0 a) Verify that this PDE is elliptic. b) Perform a change of variables on this PDE: α = ax + by β = cx + dy Choose the constants a , b , c , and d in such a way to make u αα + u ββ = 0. In other words, this PDE is just a change of basis away from Laplace’s equation. c) The cubic polynomial u ( x, y ) = x 3 - 3 xy 2 is a solution to Laplace’s equation. Determine a cubic polynomial that is a solution to u xx + u xy + u yy = 0. 10 (10 points) Eh, nothing is inspiring me. Ten free points for everyone!!!
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