Definition:Field of Directions

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Consider the following system of differential equations:

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

where $\mathbf y$ is an N-dimensional vector.

Let the boundary conditions be prescribed $\forall x\in\closedint a b$:

$\mathbf y'=\map{\boldsymbol\psi} {x,\mathbf y}$

Let these boundary conditions be consistent $\forall x_1,x_2\in\closedint a b$

Then the family of mutually consistent boundary conditions is called a field of directions for the given system $\paren{\star}$.

In other words, the first-order system is valid in an interval instead of a countable set of points.