anonymous
  • 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.
Mathematics
  • Stacey Warren - Expert brainly.com
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schrodinger
  • schrodinger
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anonymous
  • 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
  • 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
  • experimentX
do you have relevant links?

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TuringTest
  • 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
  • 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
  • experimentX
hold on for a while ... I'll work out few other probs and try to work on my copy.
anonymous
  • anonymous
oh kay. thx alot
anonymous
  • anonymous
my lecturer practically skipped this part ==
anonymous
  • anonymous
@edr1c The devil is in the not-details, in the boundary conditions. Each set of B.C. allows/disallows other solutions
anonymous
  • 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
  • anonymous
So why do we choose the coordinates ? - To fit the symmetry of the boundary of course !

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