When using separation of variables to solve a PDE the homogeneous boundary conditions are applied before the nonhomogeneous boundary or initial condition(s). Why is the order in which we apply our conditions important?
Intuitively, it seems similar to solving nonhomogenous ODE's---we find the homogeneous solution first and then find the particular solution. While I understand why techniques like variation of parameters or the method undetermined coefficients require the homogeneous solution to be found before the particular, I have not found a good source to explain the analogous case for PDE's. Maybe the ODE case is not a good analogy.
In the case of PDE's, is it that we need to make sure the separation constant (often -$\lambda^2$) is based on the general solution?
If anyone could point me in the direction of a good explanation I would really appreciate it! To be clear, I am not asking about why we need homogeneous boundary conditions; I'm asking why the homogeneous conditions need to be applied before the inhomogeneous. Thanks!