AMath350.F13.A8.Sol

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

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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) &gt; 1. Never occurs. =) u⇠⌘ = ˆ 2 64 parabolic if sin (x) = 1 ) x = 2 + k ⇡ , k Z Z hyperbolic if2sin2 (x) &gt; 1. Never occurs. elliptic if =) (x) &lt; 1 = ⌘ + g (⇠ ) sin 2 u ˆ= parabolic if sin (x) ⇠ 1 64 ) x = 2 + k ⇡ , k Z Z 2 elliptic if ⇠⌘ (x) &lt; 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...
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