Ordinary differential equations Definition

Boundary value problem

Builds on Ordinary differential equation Solution of an ordinary differential equation Initial value problem

Definition

Let $a < b$ and $n \ge 1$ , and let $F \colon [a, b] \times \R^{n} \to \R$ be continuous. Let $g \colon \R^{n} \times \R^{n} \to \R^n$ be a function. A (two-point) boundary value problem on $[a, b]$ is the problem of finding a function $y \in C^n([a, b])$ such that:

$y$ is a solution of the ordinary differential equation $y^{(n)}(x) = F\left( x, y(x), \ldots, y^{(n-1)}(x) \right)$ for all $x \in [a, b]$ , and
$y$ satisfies the boundary conditions $g\left( \mathbf{y}(a), \mathbf{y}(b) \right) = 0$ , where $\mathbf{y}(x) = \left( y(x), y'(x), \ldots, y^{(n-1)}(x) \right)$ .

The problem is linear if the equation is linear in $y, \ldots, y^{(n)}$ and $g$ is affine. A linear problem is homogeneous if $y = 0$ satisfies the equation and the conditions.

Remarks

If the conditions only prescribe $\mathbf{y}(a)$ , that is, $g(u, v) = u - \mathbf{y}_0$ for a fixed $\mathbf{y}_0 \in \R^n$ , the problem is an initial value problem, not a boundary value problem in the proper sense. The same name is used for systems and for partial differential equations on a domain, with conditions on its boundary.

Example: $y'' = -y$ on $[0, \frac{\pi}{2}]$ with $y(0) = 0$ and $y(\frac{\pi}{2}) = 1$ , that is, $g(u, v) = (u_1, v_1 - 1)$ . The function $y = \sin x$ solves the equation, $\sin 0 = 0$ and $\sin \frac{\pi}{2} = 1$ ; it is the only solution, since the general solution is $c_1 \cos x + c_2 \sin x$ .

Non-example of solvability: $y'' = -y$ on $[0, \pi]$ with $y(0) = 0$ and $y(\pi) = 1$ is a boundary value problem without solution. Every solution with $y(0) = 0$ is $c \sin x$ , which vanishes at $\pi$ .

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Initial value problem concept
Ordinary differential equation concept
Solution of an ordinary differential equation concept
Boundary value problem definition

Tag	`1HGGRS57KT`
URI	`m2://1HGGRS57KT`
Kind	definition
Writing	W0 stub
MSC	34B05, 34B15
Defines	Boundary value problem
Uses	Ordinary differential equation, Solution of an ordinary differential equation, Initial value problem
Changed	2 November 2026

Boundary value problem

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Remarks

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