Question

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.

Answer 1

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?

Answer 2

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.

Answer 3

do you have relevant links?

Answer 4

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

Answer 5

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?

Answer 6

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

Answer 7

oh kay. thx alot

Answer 8

my lecturer practically skipped this part ==

Answer 9

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

Answer 10

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

Answer 11

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