Unformatted text preview: unique solution:
, and
3 ex+x
2
u(x, order,xlinear PDEs, determine in which regions
y ) = + 2 (1 e x y ).
2. For each of the following second
x
of R2 the region is hyperbolic, parabolic, and elliptic. Sketch and label the
(iii) u(x, 0) = x2 , uy (1, y ) = e2y . As above H (x) = x2 G(x).
regions.
Apply second BC: uy (1, y ) = e2y ) e y G(1) = e2y ) G(1) = e3y .
This uxy + 2yuyy + the x x u = xyex2
a) yuxx is4impossible, so exy uBVP 3has no solutions.
2 2 5. (a) yuxx 4uxy + 2yuyy + exy ux x3u = xyex
a = y, b = 4, c = 2y ) b2 4ac = 16 8y 2 .
Equation is hyperbolic if 16 8y 2 > 0
p
Equation is parabolic if 16 8y 2 = 0 ) y = ± 2
Equation is elliptic if 16 8y 2 < 0 (b) 2uxx + cos(x)uxy + (1 sin2(x))uyy + cos2(x)u + sin(xy )ux
a = 2, b = cos(x), c = (1 sin2(x))
) b2 4ac = cos2 (x) 8(1 sin2 (x)) = 7(sin2 (x) 1). uy = tan(x) AMATH 350 Page 2 Assignment #8 Solutions  Fall 2012 b) 2uxx + cos(x)uxy + 1 sin2 (x) uyy + cos2 (x)u + sin (xy ) ux uy = tan(x) (b) 2uxx + cos(x)uxy + (1 sin2(x))uyy + cos2(x)u + sin(xy )ux
a = 2, b = cos(x), c = (1 sin2(x))
) b2 4ac = cos2 (x) 8(1 sin2 (x)) = 7(sin2 (x) 1).
Equation is hyperbolic i...
View
Full Document
 Fall '14
 Exponential Function, Sin, Complex number, B2 Spirit

Click to edit the document details