AMath350.F13.A8.Sol

# Noting that d x e 0 f c let y 2x 1 8x

This preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: rabolic, or elliptic. a) 2 (a) 1 u = 10, x x + sin(y 4 0 6. uxx a =01,xyb + 16uyy c =u16 ) b )u =ac. = 36 > 0. PDE is hyperbolic. (b) a = , b Characteristic equations: (a) Find1the = 10, c = 16 ) b2 4ac = 36 > 0. PDE is hyperbolic. b) characteristics. p p b2 4ac dy b dy (b) Characteristic equations:b + b2 4ac = = p2a p2a dx dx dy = b 8 b2 4ac dy = b + b2 4ac =2 = dx 2a dx 2a Integrating: =2 =8 y = 2 x + k1 y = 8 x + k2 Integrating: y + 2 x = k1 y + 8 x = k2 y = 2 x + k1 y = 8 x + k2 y + ⇠y = k (c) Let ⇠ = y + y x, 2x == +1 8x, then ⇠x = 2, 8x = 1, 2⌘x = 8, ⌘y = 1 and 2+ ⌘ yk all second partial derivatives are zero. Noting that d = x, e = 0, f = (c) Let ⇠ = y + 2x, ⌘ = 1 + 8x, then ⇠x = 2, ⇠y = 1, ⌘x = 8, ⌘y = 1 and y sin(y ), g = 0, and x = 6 (⌘ ⇠ ), y = 1 (4⇠ ⌘ ), we obtain g = 0 and ˆ 3 all second partial derivatives are zero. Noting that d = x, e = 0, f = 1 B 0, 2(1)(2)(8) + sin(y ), g = = and x = 6 (⌘ 10 ), y = 1 (4⇠ ⌘ ),+ 2(16)(1)(1)= 0 and ⇠ ((2)(1) 3 (8)(1)) we obtain g = 36 ˆ 1 1 1 (⌘ ⇠ )(2) 0 ⇠ ⌘ D = 2(1)(2)(8) 1=((2)(1) + (8)(1)) + 2(16)(1)(1) = 36 B 6 3 3 1 4 4 1 (⌘ ⇠ )(2) = 1 ⇠ 1 ⌘ D= (⌘ ⇠ )(8) = ⇠ ⌘ E= dx = 2a 2 dx = 8 2a Assignment #8 Solutions - Fall 2013 Page 5 Integrating: y = 2 x + k1 y = 8 x + k2 y + 2 x = k1 y + 8 x = k2 c) Put the equation in its canonical form. AMATH 350 (c) Let ⇠ = y + 2x, ⌘ = y + 8x, then ⇠x = 2, ⇠y = 1, ⌘x = 8, ⌘y = 1 and all second partial derivatives are zero. Noting that d = x, e = 0, f = 1 sin(y ), g = 0, and x = 6 (⌘ ⇠ ), y = 1 (4⇠ ⌘ ), we obtain g = 0 and ˆ 3 B = 2(1)(2)(8) 10 ((2)(1) + (8)(1)) + 2(16)(1)(1) = 1 1 1 (⌘ ⇠ )(2) = ⇠ ⌘ D= 6 3 3 4 4 1 (⌘ ⇠ )(8) = ⇠ ⌘ E= 6 3 3 4 1 ˆ f = sin(y (⇠ , ⌘ )) = sin( ⇠ ⌘) 3 3 36 1 u3 u Thus the PDE become...
View Full Document

## This note was uploaded on 02/08/2014 for the course AMATH 350 taught by Professor Davidhamsworth during the Fall '12 term at University of Waterloo, Waterloo.

Ask a homework question - tutors are online