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Hi

I'm getting confused solving the Laplace eqn in Cartesian coordinates.

The equation can be solved by solving each of

[itex]

\frac{X''(x)}{X(x)}=-k_x^2, \qquad\qquad

\frac{Y''(y)}{Y(y)}=-k_y^2, \qquad\qquad

\frac{Z''(z)}{Z(z)}=k_z^2

[/itex]

and then substituting into the equation

[itex]

\Phi(\vec{x})=X(x)Y(y)Z(z)

[/itex]

The solution to the (general) ODE's is

[itex]

X(x)=A_1e^{ik_xx}+A_2e^{-ik_xx}, \qquad\qquad

Y(y)=B_1e^{ik_yy}+B_2e^{-ik_yy}, \qquad\qquad

Z(z)=C_1e^{k_zz}+C_2e^{k_zz}

[/itex]

But the solution when solving for the electric potential is given by

[itex]

\Phi_{k_x,k_y,\pm}(\vec{x})=exp\left(ik_xx+ik_yy \pm\sqrt{k_x^2+k_y^2}\right)

[/itex]

Where am I going wrong?

Why have they let the coefficients [itex]A_{1,2}, B_{1,2}, C_{1,2}=1[/itex]?

Thanks

I'm getting confused solving the Laplace eqn in Cartesian coordinates.

The equation can be solved by solving each of

[itex]

\frac{X''(x)}{X(x)}=-k_x^2, \qquad\qquad

\frac{Y''(y)}{Y(y)}=-k_y^2, \qquad\qquad

\frac{Z''(z)}{Z(z)}=k_z^2

[/itex]

and then substituting into the equation

[itex]

\Phi(\vec{x})=X(x)Y(y)Z(z)

[/itex]

The solution to the (general) ODE's is

[itex]

X(x)=A_1e^{ik_xx}+A_2e^{-ik_xx}, \qquad\qquad

Y(y)=B_1e^{ik_yy}+B_2e^{-ik_yy}, \qquad\qquad

Z(z)=C_1e^{k_zz}+C_2e^{k_zz}

[/itex]

But the solution when solving for the electric potential is given by

[itex]

\Phi_{k_x,k_y,\pm}(\vec{x})=exp\left(ik_xx+ik_yy \pm\sqrt{k_x^2+k_y^2}\right)

[/itex]

Where am I going wrong?

Why have they let the coefficients [itex]A_{1,2}, B_{1,2}, C_{1,2}=1[/itex]?

Thanks

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