Alpha Equation
The Alpha Equation is a non-standard term and does not represent a standard mathematical equation. Without a clear definition or specific form, it is not possible to determine whether separation of variables can be applied to solve it. Separation of variables is a method commonly used to solve partial differential equations, where the variables can be separated into individual functions that are then solved separately before being combined into a general solution.

Laplace Equation
The Laplace Equation is a second-order partial differential equation that describes the behavior of harmonic functions. It can be solved using the method of separation of variables, where the solution is written as a product of functions of different variables that are then solved independently.

Helmholtz Equation
The Helmholtz Equation is also a second-order partial differential equation that arises in many areas of physics and engineering, particularly in the study of wave propagation. Like the Laplace Equation, the Helmholtz Equation can be solved using separation of variables by assuming a solution of the form of a product of functions of different variables.

Biharmonic Equation
The Biharmonic Equation is a fourth-order partial differential equation that describes bending of thin plates and other phenomena. While it is more complex than the Laplace and Helmholtz Equations, it can still be solved using separation of variables by introducing additional auxiliary variables to reduce the equation to a set of simpler equations that can be solved independently.
In conclusion, the Alpha Equation is not a standard mathematical equation, and without a specific form or definition, it is unclear whether separation of variables can be applied to solve it. However, the Laplace Equation, Helmholtz Equation, and Biharmonic Equation can all be solved using separation of variables due to their well-defined forms and properties.