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- anonymous

any1 can explain the principles in solving laplace equation in rectangular coordinates?
i know i need to solve for 2 2nd order ODE but im not sure how to conclude for whether they are trivial or nontrivial.

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- anonymous

- schrodinger

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- anonymous

i let u(x, y) = F(x) G(y)
so
Uxx = F'' G
Uyy = F G"
then the equation become F"G + FG" = 0
i separate the variables and equate them to a separation constant, k
so the 2 2nd order ODE are
F''-kF=0 , G"+kG = 0
how do i proceed with the initial conditions given?

- anonymous

i need to get 3 cases when k=0, k<0 and k>0. but i dont know how to conclude after applying the initial conditions into the cases.

- experimentX

do you have relevant links?

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- TuringTest

I always forget how to do this, so here is a hopefully helpful link
http://tutorial.math.lamar.edu/Classes/DE/LaplacesEqn.aspx

- anonymous

let say i have the case where k=0
for F"=0
i got F=Ax+B, and i substitube initial conditions to obtain F be trivial (F=0).
for G"=0
i got G=Cx+D
G(0)=0 :
D=0, G=Cx
G(24)=24:
C=1, G=x ??
is it possible to obtain nontrivial solution for G when F is trivial?

- experimentX

hold on for a while ... I'll work out few other probs and try to work on my copy.

- anonymous

oh kay. thx alot

- anonymous

my lecturer practically skipped this part ==

- anonymous

@edr1c The devil is in the not-details, in the boundary conditions.
Each set of B.C. allows/disallows other solutions

- anonymous

The coordinates is in some sense "our imagination" - the equation is not influenced by the coordinates we choose. It IS influenced by the symmetry of its BOUNDARY, and of its BOUNDARY CONDITIONS

- anonymous

So why do we choose the coordinates ? - To fit the symmetry of the boundary of course !

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