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Unformatted text preview: MthSc 208, Fall 2011 (Differential Equations) Dr. Macauley HW 21 Due Friday April 29th, 2011 (1) Which of the following functions are harmonic? (a) f ( x ) = 10 3 x . (b) f ( x,y ) = x 2 + y 2 . (c) f ( x,y ) = x 2 y 2 . (d) f ( x,y ) = e x cos y . (e) f ( x,y ) = x 3 3 xy 2 . (2) (a) Solve the following Dirichlet problem for Laplace’s equation in a square region: Find u ( x,y ), 0 ≤ x ≤ π , 0 ≤ y ≤ π such that ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = 0 , u (0 ,y ) = u ( π,y ) = 0 , u ( x, 0) = 0 , u ( x,π ) = 4sin x 3sin2 x + 2sin3 x. (b) Solve the following Dirichlet problem for Laplace’s equation in the same square region: Find u ( x,y ), 0 ≤ x ≤ π , 0 ≤ y ≤ π such that ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = 0 , u (0 ,y ) = 0 , u ( π,y ) = y ( π y ) , u ( x, 0) = u ( x,π ) = 0 (c) By adding the solutions to parts (a) and (b) together (superposition), find the solution to the Dirichlet problem: Find u ( x,y ), 0 ≤ x ≤ π , 0 ≤ y ≤ π such that ∂ 2 u ∂x 2 + ∂ 2 u ∂y 2 = 0 , u (0 ,y ) = 0 , u ( π,y ) = y ( π y ) , u ( x, 0) = 0 , u ( x,π ) = 4sin x 3sin2 x + 2sin3 x. (d) Sketch the solutions to (a), (b), and (c). Hint: it is enough to sketch the boundaries, and then use the fact that the solutions are harmonic functions ....
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This note was uploaded on 03/11/2012 for the course MTHSC 208 taught by Professor Staufeneger during the Spring '09 term at Clemson.
 Spring '09
 Staufeneger
 Differential Equations, Equations

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