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Unformatted text preview: 1 Homework #3 1. Suppose F ( x,u,p ) = u 2 p 2 , and F ( u ) = R b a F ( x,u ( x ) ,u ( x )) dx (a) Show all extremals are parabolas. Solution The Euler equation for this functional is 2 up 2 d dx (2 u 2 p ) = 0 2 up 2 (4 up 2 + 2 u 2 p ) = 0 u ( u ) 2 + u 2 u 00 = 0 u ( ( u ) 2 + uu 00 ) = 0 = ⇒ u = 0 or ( u ) 2 + uu 00 = 0 Notice that ( u 2 ) = 2 uu and ( u 2 ) 00 = 2( u ) 2 + 2 uu 00 . Therefore, either u = 0 or ( u 2 ) 00 = 0 = ⇒ u ( x ) 2 = C 1 x + C 2 (b) Find u , the extremal satisfying the boundary conditions u ( a ) = 1 and u ( b ) = 0, where a < b . Sketch a graph of the extremal. Solution With u ( a ) = 1 ,u ( b ) = 0 1 2 = C 1 a + C 2 , 2 = C 1 b + C 2 = ⇒ C 1 = 1 a b ,C 2 = b a b u ( x ) 2 = x b a b (c) Compute F ( u ) and F ( u ) where u ( x ) is the linear function satisfying u ( a ) = 1 and u ( b ) 0. Solution u ( x ) 2 = x b a b 2 u u = 1 a b = ⇒ ( u ) 2 = 1 (2 u ( a b )) 2 1 Thus F ( u ) = Z b a u 2 ( u ) 2 dx (1) = Z b a u 2 4 u 2 ( a b ) 2 dx (2) = Z b a 1 4( a b ) 2 dx (3) = 1 4( a b ) 2 x  b a (4) = 1 4( a b ) (5) For the linear function, u = x b a b F ( u ) = Z b a u 2 ( u ) 2 dx (6) = Z b a ( x b ) 2 ( a b ) 2 1 ( a b ) 2 dx...
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This note was uploaded on 09/09/2009 for the course MATH 410 taught by Professor Staff during the Spring '08 term at Maryland.
 Spring '08
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