X2 4 2y uyy 4u 4f xe 2y 4gxe2y xy 4 0 4gxe2y

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Unformatted text preview: yy + 4+x 4 + x2 2 4. uyy Find uy =general solution. a) + x the 0 @ (a) Rewrite PDE as: ( uy + x2 u) = 0 @y Integrate w.r.t. y holding x fixed: uy + x2u = F (x) 2 2 Linear DE w.r.t. y with IF: e x dy = ex y . Solving: 2 2 2 ex y uy + x2ex y u = ex y F (x) @ x2 y 2 (e u) = ex y F (x) @y Integrate w.r.t. y holding x fixed 2 ex y x2 y e u= F (x) + G(x). x2 General Solution: 1 2 u(x, y ) = 2 F (x) + e x y G(x), x 2 = H (x) + e x y G(x) where H, G are arbitrary functions. b) Verify that your solution solves the DE. (b) Di erentiating (b) Di erentiating 2 2 uy = x2e x 2y G(x), uyy = x4e x 2y G(x) 2 xy 4 xy uy = x e G(x), uyy = x e G(x) 2 2 4 x2 y Substituting into the PDE: uyy + x 2uy = x 4e x2 yG(x) x4e x 2y G(x) = 0, 4 xy Substituting into the PDE: uyy + x uy = x e G(x) x e G(x) = 0, 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 (i) u(x, 0) = x 2, uy (0, y ) = y . (i) u(x, 0) = x , u (0, y ) = y . u(x, 0) = x2 ) H (y ) + G(x)...
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