Separation of variables for a Second Order PDE with three variables I am fairly new to PDEs and would like some hints on how to approach a particular question.  
I have been given six boundary conditions for the equation below (L can be assumed to be a known constant):
$\ \partial^2 f(x,y,z)/\partial x^2 + \partial^2 f(x,y,z)/\partial y^2 + \partial^2 f(x,y,z)/\partial z^2  - 1/L^2 f(x,y,z) =0 \ $
equation
I would like to fix the analytical solution for $\ f(x,y,z) \ $, but am unsure of how to approach this kind of problem with three independent variables along with
 $\ - 1/L^2 f(x,y,z) \ $ term.  
I have only really dealt with two independent variables in the form            
$\ \partial^2 f(x,y)/\partial x^2 + \partial^2 f(x,y)/\partial y^2 =0 \ $ 
I am not sure how to deal the $\ - 1/L^2 f(x,y,z) \ $ term and would like to ask what is the best way to approach this problem.  I know I should be using the separation of variables method, but don't know the intermediate step to deal with the equation I have and get it into $\ \partial^2 f(x,y,z)/\partial x^2 + \partial^2 f(x,y,z)/\partial y^2 + \partial^2 f(x,y,z)/\partial z^2 =0 \ $
I am not entirely sure how to deal with $ \ - 1/L^2 f(x,y,z) \ $ term
Once I find out how, I plan to follow the approach outlined in 
Separation of variables with three independent variables
Thanks for reading and hope you can help!
Cheers
 A: $\frac{\partial^2 f(x,y,z)}{\partial x^2} + \frac{\partial^2 f(x,y,z)}{\partial y^2} + \frac{\partial^2 f(x,y,z)}{\partial z^2}  - \frac{1}{L^2} f(x,y,z) =0  \tag 1$
Separation of variables means that you are looking for particular solutions on a particular form :
$$f(x,y,z)=X(x)Y(y)Z(z)$$
Putting it into Eq.$(1)$ leads to :
$$X''YZ+ y''XZ+Z''XY-\frac{1}{L^2} XYZ=0$$
$$\frac{X''}{X}+\frac{Y''}{Y}+\frac{Z''}{Z}-\frac{1}{L^2}=0$$
This implies that each term is constant :
$$\begin{cases}
\frac{X''}{X}=\alpha \quad\to\quad \frac{d^2X}{dx^2}=\alpha\: X(x)\\
\frac{Y''}{Y}=\beta \quad\to\quad \frac{d^2Y}{dy^2}=\beta\: Y(y)\\
\frac{Z''}{Z}=\gamma \quad\to\quad \frac{d^2Z}{dz^2}=\gamma\: Z(z)\\
\end{cases} \qquad \text{with}\qquad \alpha+\beta+\gamma=\frac{1}{L^2}$$
Solve the three ODEs respectively for $X(x)$ , $Y(y)$ and $Z(z)$.
The parameters $\alpha$, $\beta$, $\gamma$ appear into them. So we will write them as : $X_\alpha(x)$, $Y_\beta(y)$, $Z_\gamma(z)$ 
The particular solution obtained is :
$$f(x,y,z)=X_\alpha(x) Y_\beta(y) Z_\gamma(z) \qquad \text{with}\qquad \alpha+\beta+\gamma=\frac{1}{L^2}$$
Of course this is not the general solution of Eq.$(1)$. Any linear combination of the above particular solutions is a solution of Eq.$(1)$ .
Then, all depends on the boundary conditions, in order to determine the convenient linear combination. Generally, this is the most difficult part of the task.
