What you call "combination of variables" I think is often called "separation of variables" in my humble sphere of math. Let's take an example:
$$u_{xx}+u=u_t$$
Here, $u$ is the dependent variable and $x,t$ are independent. Usually, we suppose $u(x,t) = X(x)T(t)$ hence $u_{xx} = X''T$ and $u_t = XT'$. Substituting into the PDE gives,
$$X''T+XT=XT' $$
If the solution we seek is nontrivial then division by $XT$ is allowed,
$$ \frac{X''}{X}+1 = \frac{T'}{T} $$
At this point we note that the left and right sides of this equation are functions
of variables which we took as independent. It follows there is some constant $K$ such that:
$$ \frac{X''}{X}+1 = \frac{T'}{T} =K$$
Hence, we find solving the PDE reduces to solving two ODEs:
$$ T' = KT \qquad X''+(1-K)X=0. $$
Notice that $K$ is arbitrary with the data given. This means that we don't
really have just two ODES, we have infinitely many. What typically happens is that
in a well-posed problem you are also given Boundary Values (BV) and an initial condition.
In nice problems the BVs will force $K$ to be a certain discrete set of values, countably many in fact. Then for each choice of $K$ you get two ODEs and a couple of solutions to the common BV. After all of this you sum these together to form the formal solution and match it to the initial condition(s) by the Fourier technique. So, in that sense there are not just a couple constants as in the solution to a single ODE. There is infinite flexibility in the solution because you have a family of ODEs forming the solution.
Best Answer
When we separate variables, we assume that the solution can be written as the product of functions of the different variables. Then, when focusing on the equation in one of the variables only, we often find quantized solutions, due to the boundary conditions - a simple example is the Schrodinger equation in 2D for an infinite square well. The general solution will be a weighted sum (Fourier series) of these normal modes, with the weights being the arbitrary constants we index by n. If the spectrum of modes is continuous (for example, the Schrodinger equation for a particle in free space), we take a Fourier transform, with the weight being a function of whatever parameter is involved. If we then construct the full solution, each function (whether expanded in terms of its Fourier modes or not) will be multiplied by whatever function of the other variables was found when solving the relevant equations. Since we have solved the equation in all of the variables and already imposed the BCs in the process, those functions are determined.
The solutions appear different because the boundary (or initial) conditions are imposed at different stages of the process.