From the previous chapter we have the Laplace's differential operator in ultraspherical coordinates:

Where we have the recursive operator, with indices: m=2,..,n-1

Now consider the differential equation:

Supposing that a separable solution can be expressed in the form

we will make an attempt to separate the arguments in equation (3.3). First we will separate the radial coordinate r . Denote for brevity:

consequently

Now we can separate the argument r , we denote the separation constant qn-1:

The first one of the separated equations is the radial equation

The rest, to be separated further

We can rewrite the equation (3.13) by means of the recursion formula for W.

We can separate here the argument of by using:

Moving the argument to be separated, the one with the highest index, to the left side

We denote now the separation constant q_n-2 and get the two separated equations

First of he two equations is now fully separated in the highest index angular argument, and becomes

The other one which still contains non separated angular arguments

The equation (3.22) is of the same form, but one lower index value, as equation (3.13). In a similar way we can separate all of the arguments. So we proceed and separate the angular argument with the index m in the corresponding angular differential equation:

where we assume that the solution has the form

Substituting (3.24) into (3.2) we get

We will now separate q_m , denote the separation constant q _m-1 , then for m=2,...,n-1

We got rid of the recursive expressions. The recursivity, though, is left in the total solution given in (3.4) and (3.24), but because it is simply a product of the solutions in the different separated arguments, it is easy to handle. Note, that we got the angular solution by multiplicating the lower dimensional total angular solution with a new separated function of one higher dimension. The separated angular solutions are not dependent of the actual number of dimensions of the total problem, whereas the radial solution is. Finally, all the differential equations gathered in one place:

The Radial Equation, in n-dimensional spherical coordinates

The Angular Equation, number m in order, m = 2,..,n-1

The Azimuth Equation, the first one of the angular equations

Now, all we need is use standard methods to get a numerical solutions to the differntial equations.
But why not try to analyze the equations a bit further? That's what is done in the following chapters.