410HW2 - Math 410 Homework Solutions February 4, 2009 1...

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Unformatted text preview: Math 410 Homework Solutions February 4, 2009 1 Homework #2 Find the extremal for the variational integral F ( u ) = Z b a F ( x, u ( x ) , u ( x )) dx subject to the boundary conditions u ( a ) = c, u ( b ) = d . 1. F ( x, u, p ) = 2 u + p 2 , ( a, c ) = (0 , 0) and ( b, d ) = (1 , 1). F u- d dx F p = 0 = (2)- d dx (2 p ) = 0 = 2- 2 u 00 = 0 = u 00 = 1 The general solution to this differential equation is u ( x ) = 1 2 x 2 + C 1 x + C 2 Using boundary conditions, u (0) = 0 = C 2 = 0, and u (1) = 1 , = C 1 + 1 / 2 = 1 . Therefore, u ( x ) = 1 2 ( x 2 + x ) . 2. F ( x, u, p ) = p 2 + 2 up, ( a, c ) = (- 1 , 1) and ( b, d ) = (2 , 0). F u- d dx F p = 0 = 2 p- d dx (2 p + 2 u ) = 0 1 = 2 u- (2 u 00 + 2 u ) = 0 = u 00 = 0 The general solution is u ( x ) = C 1 x + C 2 Using boundary conditions, u (- 1) = 1 = C 2- C 1 = 1 , u (2) = 0 = 2 C 1 + C 2 = 0. Together, this implies u ( x ) = 1 3 (2- x. ) 3. F ( x, u, p ) = p 2 + 2 xp + x 2 , ( a, c ) = (0 , 0) and (...
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410HW2 - Math 410 Homework Solutions February 4, 2009 1...

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