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A boundary condition prescribes values of an unknown function or its derivatives at the boundary of the domain of a differential equation.

The general solution of a differential equation is a family containing every solution, usually parameterized by arbitrary constants or initial data. For a forced linear ordinary differential equation, it is the sum of one particular solution and the general homogeneous solution.

An integral representation expresses a function through an integral of specified data and an integral kernel. Green-function representations solve differential equations this way, while the Bromwich inversion formula reconstructs a function from its Laplace transform.

For fixed real $\nu$ as $z\to +\infty$, the Modified Bessel function of the first kind and Modified Bessel function of the second kind have asymptotic expansionsThe expansion and its error bounds establish actual growing and decaying solutions, beyond formal matching of a differential equation. These formulas are documented in NIST DLMF, equations 10.40.1–2.

The differential equation defining modified Bessel functions of order $\nu$. For real fixed $\nu$ and positive $z$, its independent standard solutions are $I_{\nu }(z)$ and $K_{\nu }(z)$.

Choice of a control input to minimize an objective subject to differential equations for a state and constraints on the control, state or terminal time. Necessary conditions such as the Pontryagin maximum principle require regularity hypotheses and do not by themselves prove global optimality.