This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 (...
View
Full
Document
 Spring '08
 staff
 Math

Click to edit the document details