Definition: A second-order boundary value problem in the (unknown) variable \( y(x) \) on \( [L, R] \) is a system of the form:

\[ \begin{cases} -(p(x) y'(x))' + r(x) y'(x) + q(x) y(x) = f(x), & L \leq x \leq R, \\[8pt] a_L y(L) + b_L y'(L) = c_L, & \\[8pt] a_R y(R) + b_R y'(R) = c_R, & \end{cases} \]

where the last two equations are called boundary conditions and all constants and functions are known.

You may assume \( p(x) > 0 \) and \( q(x) \geq 0 \).

Definition: Assume a boundary condition of the form:

\[ a_L y(L) + b_L y'(L) = c_L, \]

The classification of the boundary condition depends on the values of \( a_L \) and \( b_L \):

If \( a_L \neq 0, b_L = 0 \), we call it a Dirichlet boundary condition.

If \( a_L = 0, b_L \neq 0 \), we call it a Neumann boundary condition.

If \( a_L \neq 0, b_L \neq 0 \), we call it a Robin or mixed boundary condition.