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Unformatted text preview: Homework 3 PHZ 5156 Due Tuesday, September 22, 2009 1. Consider the diffusion or heat flow equation in two spatial dimensions ∂ 2 ∂x 2 + ∂ 2 ∂y 2 u = 1 α 2 ∂u ∂t a) Use the method of separation of variables and take u ( x,y,t ) = F ( x,y ) T ( t )to show that this equation can be reduced to two ordinary differential equations ∂ 2 ∂x 2 + ∂ 2 ∂y 2 F ( x,y ) + k 2 F ( x,y ) = 0 and dT dt = k 2 α 2 T b) Find the general set of solutions subject to the boundary conditions u ( x = ,y,t ) = 0, u ( x = L,y,t ) = 0, u ( x,y = 0 ,t ) = 0, u ( x,y = L,t ) = 0 (Hint: You might want to start by first applying separation of variables to the spatial equation, taking F ( x,y ) = X ( x ) Y ( y )). 2. The gravitational potential Φ (potential energy per unit mass) at a distance r away from the center of the Earth is given by, Φ( r ) = GM r where r = ( x 2 + y 2 + z 2 ) 1 2 , and G = 6 . 67 × 10 11 Nm 2 kg 2 , and the mass of the Earth is M = 5 . 97 × 10 24 kg ....
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 Fall '08
 Johnson,M
 Mass, Planet, ORDINARY DIFFERENTIAL EQUATIONS

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