410HW2

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

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
This is the end of the preview. Sign up to access the rest of the 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

{[ snackBarMessage ]}

### Page1 / 5

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online