U u u u u u u amath 350 page 4 assignment 8 solutions

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: = u ⇠ ⇠ y + u ⌘ ⌘y = u ⇠ ˆ ˆ ˆ uxx = u⌘ . ˆ @ @ ( ux ) = (3ˆ⇠ + 5ˆ⌘ ) = 3 (ˆ⇠⇠ ⇠x + u⇠⌘ ⌘x ) + 5 (ˆ⌘⇠ ⇠x + u⌘⌘ ⌘x ) u u u ˆ u ˆ @x @x = 3 (3ˆ⇠⇠ + 5ˆ⇠⌘ ) + 5 (3ˆ⌘⇠ + 5ˆ⌘⌘ ) = 9ˆ⇠⇠ + 30ˆ⇠⌘ + 25ˆ⌘⌘ . u u u u u u u AMATH 350 Page 4 Assignment #8 Solutions - Fall 2013 uxy = @ @ ( ux ) = (3ˆ⇠ + 5ˆ⌘ ) = 3 (ˆ⇠⇠ ⇠y + u⇠⌘ ⌘y ) + 5 (ˆ⌘⇠ ⇠y + u⌘⌘ ⌘y ) u u u ˆ u ˆ @y @y = 3 (ˆ⇠⇠ u uyy = u⇠⌘ ) + 5 (ˆ⌘⇠ ˆ u @ @ ( uy ) = (ˆ⇠ u @y @y = (ˆ⇠⇠ u u⇠⌘ ) ˆ u⌘⌘ ) = 3ˆ⇠⇠ + 2ˆ⇠⌘ ˆ u u u⌘ ) = (ˆ⇠⇠ ⇠y + u⇠⌘ ⌘y ) ˆ u ˆ (u⌘⇠ ˆ u⌘⌘ ) = u⇠⇠ ˆ ˆ 5u⌘⌘ . ˆ (u⌘⇠ ⇠⌘ + u⌘⌘ ⌘y ) ˆ ˆ 2u⇠⌘ + u⌘⌘ . ˆ ˆ We’re now ready to rewrite the given PDE, and integrate: uxx + 2uxy =) 15uyy = 1 (9ˆ⇠⇠ + 30ˆ⇠⌘ + 25ˆ⌘⌘ )+2 (3ˆ⇠⇠ + 2ˆ⇠⌘ u u u u u =) 5u⌘⌘ ) 15 (ˆ⇠⇠ ˆ u 2u⇠⌘ + u⌘⌘ ) = 1 ˆ ˆ 64ˆ⇠⌘ = 1 u 1 hyperbolic if sin2 (x) > 1. Never occurs. =) u⇠⌘ = ˆ 2 64 parabolic if sin (x) = 1 ) x = 2 + k ⇡ , k Z Z hyperbolic if2sin2 (x) > 1. Never occurs. elliptic if =) (x) < 1 = ⌘ + g (⇠ ) sin 2 u ˆ= parabolic if sin (x) ⇠ 1 64 ) x = 2 + k ⇡ , k Z Z 2 elliptic if ⇠⌘ (x) < 1 sin u( ⇠ , ⌘ ) = ˆ + G( ⇠ ) + H ( ⌘ ) (where G0 (⇠ ) = g (⇠ )) 64 Returning to the original variables, the general solution is Equation Equation Equation Equation Equation Equation =) is is is is is is u(x, y ) = 1 (3x + y ) (5x 64 y ) + G(3x + y ) + H (5x y ), where G and H are arbitrary functions (of one variable). 5. uxx 10uxy + 16uyy xux + sin(y )u = 0. 6. uxx Classify the equationuas+ sin(y )u = 0. 10uxy + 16uyy x x hyperbolic, pa...
View Full Document

Ask a homework question - tutors are online