# Definition:Mutually Consistent Boundary Conditions

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## Definition

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

Consider the system of differential equations:

- $(1): \quad \mathbf y'' = \map {\mathbf f} {x, \mathbf y, \mathbf y'}$

Let derivatives of $\mathbf y$ satisfy:

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

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

If every solution of $(1)$ 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