Rt y with if e x dy ex y solving linear de wrt y with

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Unformatted text preview: u ex y uy + x2 ex y u y @ x22y @ x2 y (e x yu) ( e u) @y @y Integrate w.r.t. y holding x fixed Integrate w.r.t. y holding x fixed 2 2 2 xy = ex y F (x) = ex y F (x) 2 2 2 xy = ex y F (x) = ex y F (x) 2 2 xy ex 2y ex y e u= F (x) + G(x).. e u= F (x) + G(x) 2 x2 x2 General Solution: General Solution: 1 1 2 x2 y u(x, y ) = 2 F (x) + e x 2y G(x),, u(x, y ) = 2 F (x) + e x y G(x) x2 x 2 x2 y = H (x) + e x 2y G(x) = H (x) + e x y G(x) x2 y x2 y x2 y where H, G are arbitrary functions. where H, G are arbitrary functions. b) Verify that your solution solves the DE. that your solution solves the DE. b) Verify your solution: (b) Di erentiating (b) Di erentiating (b) Di erentiating (b) Di erentiating 2 2 2 2 2 x2 y 4 x2 y uy = x22e xx22yyG(x),, uyy = x44e xx22yyG(x) uy = x22e x 2yyG(x) uyy = x44e x 2yyG(x) y yy 2 xy 4 xy x G(x), x G(x) uy = x e uyy = x e uy = x e G(x), uyy = x e G(x) y yy 2 2 2 2 4 x2 y 4 x2 y 2u = x4 e x 22yG(x) 4 e x 2y G(x) Substituting into the PDE: uyy + x 2uy = x4 x 2y x44e xx22yyG(x) = 0, Substituting into the PDE: uyy + x 2 y x4 e x 2y G(x) = 0, Substituting into the PDE: uyy + x 2uy = x 4e x yG(x) x4e x 2y G(x) = 0, Substituting into the PDE: uyy + x uy = x4e x y G(x) x4 e x y G(x) = 0, yy y yy y shows that u(x, y ) is a solution. shows that u(x, y ) is a solution. shows that u(x, y ) is a solution. shows that u(x, y ) is a solution. (c) Apply BCs. (c) Apply BCs. c) Apply BCs. 2 c) Apply BCs. (c) ApplyBCs: 2 (c) Apply BCs. 2 BCs. (i) u(x, 0) = x 2,, uy (0,, y ) = y .. (i) u(x, 0) = x 2 uy (0 y ) = y (i) u(x, 0) = x 2,, uy (0,, y ) = y .. 2 (i) u(x, 0) = x uy (0 y ) = y y 2 2 u(x, 0) = x22 ) H (y ) + G(x) = x 22 ) H (x) = x22 G(x). u(x, 0) = x22 ) H (x ) + G(x) = x2 ) H (x) = x22 G(x). x u(x, 0) = x2 ) H (x) + G(x) = x2 ) H (x) = x2 G(x). u(x, 0) = x ) H (x) + G(x) = x2 ) H (x) = x G(x). Apply second BC: uy (0,, y ) = y ) 0 = y ,, which is a contradiction. Apply second BC: uy (0 y ) = y ) 0 = y which is a co...
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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.

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