Unformatted text preview: e complementary function:
4u = sin(xy ).
a)
(a) Associated homogeneous equation: uyy 4u = 0.
Characteristic equation: 2 4 = 0 )
= 2, +2.
General solution: uh (x, y ) = F (x)e 2y + G(x)e2y (F, G arbitrary functions).
b) Find a particular solution: (b) Let up(x, y ) = M1 (x) sin(xy ) + M2 (x) cos(xy ). Derivatives:
@ up
(b) Let up(x, y ) = M1 (x=sin(xy ) x) cos(xy ) xM2 (x) sin(xy )
) xM1( + M2 (x) cos(xy ). Derivatives:
@y
@ up
@ 2up = xM1(x) cos(xy ) xM2 (x) sin(xy )
2
@ y = x M1 (x) sin(xy ) x M2 (x) cos(xy )
@2 2
y
@ up
2
2
=
Substitute into the PDE: x M1 (x) sin(xy ) x M2 (x) cos(xy )
@ y2
2
x2M1 (x) into the PDE:
Substitutesin(xy ) x M2 (x) cos(xy ) 4M1(x) sin(xy ) 4M2(x) cos(xy ) = sin(xy ). uyy
uyy Equating ) sin(xy ) x2Mlike)terms: ) 4M1(x) sin(xy ) 4M2(x) cos(xy ) = sin(xy ).
x2M1 (x coe cients of 2 (x cos(xy
1
Equating coe ) cients of (4+x2)M2(x) = 0, ) M1 =
like terms:
, M2 = 0.
(4+x2)M1(x = 1,
4 + x2
1
, M2 = 0.
(4+x2)M1 ) = 1,1
(4+ ) 2
Thus up(x, y(x) = 4+x2 sin(xyx. )M2(x) = 0, ) M1 =
4 + x2
(c) The general solution1 of the nonhomogeneous equation is
c) Form the x, y ) = 2solution, and. verify that it is correct:
Thus up( general
sin(xy )
1
u(x, y ) = F (x)e y 4+xG(x)e2y 4+x2 sin(xy ). Di erentiating gives
+2
(c) The general solution of the nonhomogeneous equation is
x
1
u(x, y ) = F (xue 2y + G(x(x2y 2y4+x2 G(xxy2y Di erentiating )
) y = 2F )e )e + 2sin( )e ).
cos(xy gives
4 + x2
2x
= 2F )e 2 2y 4G G)e )e2y x sin(xy )
uuy = 4F (x(x)ey + + 2(x(x2y + 4 +2x2 cos(xy )
yy
4 +2
x
x
2y
2y
sin(xy
uyy = PDE e + 4G( u e +
Substituting into the 4F (x)shows thatx)(x, y ) is a solution:)
4 + x2
2
Substituting (x)e y + x sin(xy ) 4 u x, is 4 solution: 4 sin(xy ) = sin(xy ).
4u = 4F (x)e 2y +4Ginto 2the PDE2 shows thatF ((x)ey )2y a G(x)e2y +
4 +2
x
4 + x2
x
4
4u uyy4+ (x)ey 2y +4G(x)e2y +
sin(xy ) 4uF (x)e 2y 4G(x)e2y +
4=
sin(xy ) = sin(xy ).
4. = F x2u = 0
4 + x2 uyy
4 + x2
@
2
4. (a) 4F x2uye = y0 G (x) e2y uy + x2u) =(0 ) 4F (x) e 2y 4G (x) e2y + 4 sin (xy )
uyy Rewrite PDE as:
+ (x) 2 +4
(+ x sin xy
@ y 4 + x2
4 + x2
In...
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 Fall '12
 DavidHamsworth
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