AMath350.F13.A7.Sol

# 4 f x2u 0 4 x2 uyy 4 x2 2 4 a 4f x2uye y0 g

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Unformatted text preview: tegratePDE as: holding xxﬁxed: 0 y + x2u = F (x) w.r.t. y @ (u + 2u) = u (a) Rewrite y =2 sin (xy )2 . y Linear DE w.r.t. y@with IF: e x dy = ex y . Solving: Integrate w.r.t. y holding x ﬁxed: uy 2 x2u = F2 (x) + 2 ex y uy x2 dy 2= ex2 y .= ex y F (x) + x ex y u Solving: Linear DE w.r.t. y with IF: e @ x2 y2 2 x22y ex y uy @ y xe ex uu = e x yF (x) + ( 2 y ) = e F (x) @ x2 y Integrate w.r.t. y holding xeﬁxed = ex2 y F (x) ( u) 2 @y ex y x2 y F (x) + G(x). Integrate w.r.t. y holding x e u = ﬁxed x2 y x2 2 General Solution: = e F (x) + G(x). ex y u 12 2 u(x, y ) = xF (x) + e x y G(x), General Solution: x2 1 x2 y 2 u(x, y ) = H2 F)(x) e e Gy G)(x), = (x + + x (x x 2 where H, G are arbitrary functions. = H (x) + e x y G(x) where H, G are arbitrary functions. 4+x 2 x2 x2 uyy = 4F (x Solutions -- -xFall 2013 uyy = 4 (#7e + 4 x yy )e 2013 2 sin(xy ) Page 7 AMATH 350 AssignmentF#7)SolutionsG(Winter 2013 2 sin(xy ) Page 6 AMATH 350 Assignment #7 )e Solutions Winter+ 6 AMATH Assignment 4 + x2 4+x Substituting into the PDE shows that u(x, y ) is a solution: Substituting into the PDE shows that u(x, y ) is a solution: 2 7. Consider the PDE 7. x2 4 x2 4 2y 2y 2y 2y 2y 2y 2y+4G(x)e2y + 2y + 2 uyy 4u = 4F (x)e +4G(x)e xyu 4 0 ( uyy 4u = 4F (x)e 40 4G(x)e2y + sin(xy ) = sin(xy ) yy u2yysin(x2 uyy = F.. x)e 2y 4G(x)e 2y+ 4 + x2 sin(xy ) = sin(xy ).. u2 + xxy )y = 0 ( uyysin(x22u)y = F..x)e yy + 2+ 2 yy 4+x 4+x 4 + x2 (6) 2y 2y 2y 2y + 4G(x)e2y + + 4G(x)e2y + 4. uyy Find2uy =general solution. 4. a) Find uy =general solution: uyy + x2 the 0 a) + x2 the 0 solution. yy y @ @ 2 ( uy + x2 u) = 0 (uy + x2 u) = 0 y @y @y 2 Integrate w.r.t. y holding x ﬁxed: uy + x2u = F (x) Integrate w.r.t. y holding x ﬁxed: uy + x2 u = F (x) y 2 2 2 2 2 dy 2y x x Linear DE w.r.t. y with IF: e x dy = ex y.. Solving: Linear DE w.r.t. y with IF: e x dy = ex y Solving: (a) Rewrite PDE as: (a) Rewrite PDE as: 2 2 2 2 2 2 xy 2 xy ex y uy + x2ex y...
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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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