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somethingstra
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Homework Statement
Assume we are in the open first quadrant in the (x,y) plane
Say we have u(x,y) a C1 function in the closed first quadrant that satisfies the PDE:
[tex]u_{y}=3u_{x}[/tex] in the open first quadrant
Boundary Conditions:
u(0,y)=0 for t greater than or equal to 0
u(x,0)= g(x) for x greater than or equal to 0
and g(x) = 0 for [tex]0\preceq x\preceq1[/tex]
g(x) = (x-1)^5 for x greater than or equal to 1
Is there a solution? And if there is, is the the only solution?
Homework Equations
For first order PDE: solution is f(Ay-Bx) where the PDE is AUx + BUy = 0
The Attempt at a Solution
[tex]u(x,y) = f(-x-3y)[/tex]
so that means
[tex]u(0,y) = f(-3y)=0[/tex]
[tex]u(x,0) = f(-x)=g(x)[/tex]
set -x = w
f(w) = g(-x)
therefore f(-x-3y) = g(x+3y)
and u(x,y) = g(x+3y)
This solution works for u(x,0) but I can't find it to work with u(0,y), if y > 1/3, then the solution would be (x-1)^5 with x greater than or equal to 1, which does not satisfy u(0,y) = 0
so I concluded that there is no solution...this doesn't seem right for me. Any help? I feel that I don't actually have to try to find u in order to determine if a solution exists...
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