In the assumption of separable form for φ, what expression is used?

The assumption of separable form for the function φ in three dimensions typically implies that the function can be represented as the product of three separate functions, each depending on one spatial variable only. This approach is often used to simplify the solving of partial differential equations, especially in contexts such as quantum mechanics or heat conduction problems.

In this context, the expression φ(x,y,z) = X(x)Y(y)Z(z) clearly illustrates this separability. Each function—X, Y, and Z—depends solely on one of the coordinates, allowing one to analyze the behavior of the overall function by examining these independent components. This method facilitates the separation of variables, making it possible to solve complex problems using simpler one-dimensional equations sequentially.

The other choices do not exhibit this characteristic of separability. For instance, the second option suggests a simple addition of constants, which does not reflect a functional dependency on the variables x, y, and z. The third option represents an exponential function dependent on the combination of all variables, not maintaining the independence of each coordinate as required for separability. Lastly, the fourth choice involves sine functions that could potentially be separable; however, its form is not final without directly expressing it in the product format as the first