4 x2 2 uyy 4uf xe 4 x2 a rewrite pde as uy

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Unformatted text preview: xed: uy +F (x)= 2y (x)G(x)e2y + 4 sin (xy ) + ( 2 ) y 2y +4G( y holding 0 Integrate sin (xy ) 4 x u e F 4 2 2 2 4 + x2 Linear DE w.r.t. y @ 4 + x e x dy = ex y . Solving: with IF: 2 (a) Rewrite PDE as: (uy + x u= = 0(xy ) . ) sin 2 2 @y x2 y e uy + x2ex y u 2 = ex y F (x) Integrate w.r.t. y holding x fixed: uy + x u = F (x) 2 @ x2 2 2 Linear DE w.r.t. y with IF: e x dy e= yex)y .= ex y F (x) ( u Solving: @y x2 y 2 x2 y x2 y Integrate w.r.t. y e uy + x fixed = e F (x) holding e u x2 y @ x22y 2 (e x yu) = e x y F (x) + G(x). = e F (x) eu @y x2 Integrate w.r.t. General Solution: y holding x fixed 1x2 y 2 (x) ) e x y G( u(ex2yu = e2 F F (x++ G(x). x), x, y ) = xx2 x2 y G(x) General Solution: = H (x) + e 1 x2 y G(x), u where H, G are arbitrary functions. (x, y ) = 2 F (x) + e x 2 = H (x) + e x y G(x) where H, G are arbitrary functions. x2 2y sin(xy ) Page 7 uyy = 4F (x)e Solutions (-xFally 2012 + 4G )e2 + AMATH 350 Assignment #7 4 + x2 Substituting into the PDE shows that u(x, y ) is a solution: 7. x2 4 2y uyy 4u = 4F (x)e 2y +4G(x)e2y + xy ) 4 0 ( 4G(x)e2y + sin(xy ) = sin(xy ). u2 sin(x2 uy =F . x)e...
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This document was uploaded on 02/09/2014.

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