Solving the Laplace equation in Cartesian Coordinates leads to the 2nd order ODEs:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]

\frac{X''}{X}=k_1, \qquad \frac{Y''}{Y}=k_2 \qquad \frac{Z''}{Z}=k_3

[/itex]

In each case the sign of [itex]k_i[/itex] will determine if the solution (to the particular ODE) is harmonic or not.

Hence, if two people solve the Laplace equation, one may get a solution that is harmonic in [itex]x[/itex], and the other harmonic in [itex]z[/itex].

How could this be?

Does this simply mean that both solutions satisfy the Laplace equation, and hence the solution is not unique?

If we have non-uniqueness then when considering an actual physical situation (such as the potential of an electric field) how can we be sure that the solution we get is the true solution?

Thanks

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# Laplace Equation in Cartesian Coor.

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