Why do Sturm-Liouville problems appear when solving PDEs by separation of variables?

Dirichlet boundary conditions correspond to setting the derivative coefficients to zero ($\alpha_2 = \beta_2 = 0$), leaving conditions on the values of $y$ at the endpoints. Neumann boundary conditions correspond to setting the function coefficients to zero ($\alpha_1 = \beta_1 = 0$), leaving conditions on the derivatives.

Conditions: The general boundary conditions are $\alpha_1 y(a) + \alpha_2 y'(a) = 0$ and $\beta_1 y(b) + \beta_2 y'(b) = 0$.; Dirichlet: $\alpha_2 = \beta_2 = 0$.; Neumann: $\alpha_1 = \beta_1 = 0$.

A Sturm-Liouville problem is regular exactly when its domain is the finite closed interval $X=[a,b]$ and the coefficient $p$ does not vanish anywhere on $X$ ($p(x) \neq 0$ for all $x \in X$). A problem is singular when either the domain $X$ is infinite/unbounded or the coefficient $p$ vanishes at some point $x_0$ in $X$.

Conditions: Regular: $X=[a,b]$ and $p(x) \neq 0$ for all $x \in X$.; Singular: $X$ is infinite, or $p(x_0)=0$ for some $x_0 \in X$.; For unbounded $X$, boundary conditions are interpreted using limits.

For real constants, the condition $\alpha_1^2 + \alpha_2^2 \neq 0$ is a compact way to state that $\alpha_1$ and $\alpha_2$ are not both zero. This ensures that the boundary condition at the endpoint $x=a$ is nontrivial.

Conditions: The constants $\alpha_1, \alpha_2$ are real numbers.; The boundary condition is $\alpha_1 y(a) + \alpha_2 y'(a) = 0$.

The Sturm-Liouville operator is the second-order differential expression $L y = (p y')' + q y$. The coefficient function $p$ must be continuously differentiable, meaning $p \in C^1(X)$, and the coefficient function $q$ must be continuous, meaning $q \in C(X)$.

Conditions: The operator acts on a function $y$.; $p$ must be continuously differentiable.; $q$ must be continuous.

A Sturm-Liouville problem is an eigenvalue problem of the form $L y = \lambda y$ on an interval $a < x < b$, together with homogeneous linear boundary conditions at the endpoints: $\alpha_1 y(a) + \alpha_2 y'(a) = 0$ and $\beta_1 y(b) + \beta_2 y'(b) = 0$. The coefficients at each endpoint cannot both be zero.

Conditions: The equation is posed on an interval $a < x < b$.; Boundary conditions are homogeneous linear combinations of $y$ and $y'$ at $x=a$ and $x=b$.; At each endpoint, the pair of boundary coefficients is not both zero.