Separation of Variables: Why the Need to Apply Homogenous Conditions First?

Question

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!

Answer 1

When you look at one of the separated ODEs in Sturm-Liouville form, $$ \frac{1}{w}\left[-\frac{d}{dx}\left(p\frac{df}{dx}\right)+qf\right] = \lambda f $$ you can fully "solve" the equation for an orthogonal basis of functions if there are endpoint conditions that result in a selfadjoint problem. A regular problem where $p$, $w$ do not vanish at the endpoints, for example, results in a sequence of eigenvalues $$ \lambda_1 < \lambda_2 < \cdots $$ and corresponding eigenfunction solutions $X_n(x)$ corresponding to the $\lambda_n$. If you do not have homogeneous conditions at both endpoints, then the $\lambda_n$ are not determined, and the $X_n$ are not determined.

If you can solve all but the last ODE in, say, $x_N$, then you can expand a general function in terms of these using unknown coefficient functions $C_{n_1,n_2,\cdots,x_{N-1}}(x_N)$: $$ u(x_1,x_2,\cdots,x_N) = \sum_{n_1}\sum_{n_2}\cdots\sum_{n_{N-1}}C_{n_1,n_2,\cdots,n_{N-1}}(x_N)X_{n_1}(x_1)X_{n_2}(x_2)\cdots X_{n_{N-1}}(x_{N-1}). $$ When you plug this proposed solution back into the PDE, you end up with ODEs to solve for the coefficents $C$. There will be homogeneous conditions at, say, the left endpoint of the $x_N$ variable, and a more general condition at the right endpoint of the $x_N$ variable chosen to match the final boundary condition of the PDE at the fixed right endpoint value $x_N$, which can be matached because a general function $F(x_1,x_2,\cdots,x_{N-1})$ can be expanded in the basis functions obtained from the $N-1$ equations where there are two homogeneous endpoint conditions.