Partial differential equation

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Jncik
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Homework Statement


solve

ux = 2*uy when u(0,y) = e^(-y)


The Attempt at a Solution



this my attempt

using separation of variables I get

X'(x) - 2 λ X(x) = 0
Y'(y) - 2 λ Y(y) = 0

now I have to figure out for different λs what's happening

so

for λ = 0 I get X'(x) = 0 hence X(x) = A where A is a constant

also Y'(y) = 0 hence Y(y) = B where B is a constant

this gives us a solution C = BA where u(x,y) = C

now from the boundary conditions we have u(0,y) = e^(-y) => C = e^(-y)

hence u(x,y) = e^(-y)

for λ>0 I get a solution X(x) = A e^(λ x) and Y(y) = B e^(λx)

from the boundary conditions I get AB = e^(-y) hence

u(x,t) = e^(λx - y)

for λ<0 I have the same thing as for λ>0 but a solution u(x,t) = e^(-y) * e^(-3 λ x)

is this correct?
 

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