# Definition:Mutually Consistent Boundary Conditions

## Definition

Let $\map {\mathbf y} x$, $\map {\boldsymbol\psi} {\mathbf y}$ be an N-dimensional vectors.

Consider the differential equations:

- $\mathbf y''=\map {\mathbf f} {x,\mathbf y,\mathbf y'}\paren{\star}$

Let derivatives of $\mathbf y$ satisfy:

- $\mathbf y'\vert_{x=x_1}=\map{\boldsymbol\psi^{\paren 1} } {\mathbf y}\vert_{x=x_1}$

- $\mathbf y'\vert_{x=x_2}=\map{\boldsymbol\psi^{\paren 2} } {\mathbf y}\vert_{x=x_2}$

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

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 6.31$: Consistent Boundary Conditions. General Definition of a Field