Equations must be compatible with their boundary conditions

A system described by the second order linear and homogeneous differential equation:

y" + p(x) y` + q(x) y = r(x)

is not compatible with just any reasonable boundary conditions, for instance the linear and inhomogeneous
boundary conditions at the end points of the interval [0,1]:

a1 y(0) + a2 y`(0) = c

and

b1 y(1) + b2 y`(1) = d

Equations and boundary conditions must not only be reasonable models independently they must also be compatible with one another.