Definition:Mutually Consistent Boundary Conditions

Definition
Let $ \mathbf y \left ( { x } \right ) $, $ \boldsymbol \psi \left ( { \mathbf y } \right ) $ be an N-dimensional vectors.

Consider the following system of differential equations:


 * $ \mathbf y'' = \mathbf f \left ( { x, \mathbf y, \mathbf y' } \right ) \left ( { \star } \right ) $

Let derivatives of $ \mathbf y $ satisfy:


 * $ \mathbf y' \vert_{ x = x_1 } = \boldsymbol \psi^{ \left ( { 1 } \right )} \left ( { \mathbf y } \right ) \vert_{ x = x_1 } $


 * $ \mathbf y' \vert_{ x = x_2 } = \boldsymbol \psi^{ \left ( { 2 } \right )} \left ( { \mathbf y } \right ) \vert_{ x = x_2 } $

If every solution of $ \left ( { \star } \right ) $ satisfying conditions at $ x = x_1 $ automatically satisfies conditions at $ x = x_2 $ (or vice versa), then these boundary conditions are called mutually consistent.