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  "body": "\n## Definition\n\nLet $a \u003c b$ and $n \\ge 1$, and let $F \\colon [a, b] \\times \\R^{n} \\to \\R$ be continuous.\nLet $g \\colon \\R^{n} \\times \\R^{n} \\to \\R^n$ be a function.\nA (two-point) *boundary value problem* on $[a, b]$ is the problem of finding a function $y \\in C^n([a, b])$ such that:\n\n1. $y$ is a [[solution|07VX34QXAV]] of the [[ordinary differential equation|0NT64JV96A]] $y^{(n)}(x) = F\\left( x, y(x), \\ldots, y^{(n-1)}(x) \\right)$ for all $x \\in [a, b]$, and\n2. $y$ satisfies the boundary conditions $g\\left( \\mathbf{y}(a), \\mathbf{y}(b) \\right) = 0$, where $\\mathbf{y}(x) = \\left( y(x), y'(x), \\ldots, y^{(n-1)}(x) \\right)$.\n\nThe problem is *linear* if the equation is linear in $y, \\ldots, y^{(n)}$ and $g$ is affine.\nA linear problem is *homogeneous* if $y = 0$ satisfies the equation and the conditions.\n\n## Remarks\n\nIf 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|0FXDW992DS]], not a boundary value problem in the proper sense.\nThe same name is used for systems and for partial differential equations on a domain, with conditions on its boundary.\n\nExample: $y'' = -y$ on $[0, \\frac{\\pi}{2}]$ with $y(0) = 0$ and $y(\\frac{\\pi}{2}) = 1$, that is, $g(u, v) = (u_1, v_1 - 1)$.\nThe 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 $c_1 \\cos x + c_2 \\sin x$.\n\nNon-example of solvability: $y'' = -y$ on $[0, \\pi]$ with $y(0) = 0$ and $y(\\pi) = 1$ is a boundary value problem without solution.\nEvery solution with $y(0) = 0$ is $c \\sin x$, which vanishes at $\\pi$.\n",
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