{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "1HGGRS57KT"
    },
    {
      "field": "uses",
      "target": "7P72CCAMGZ"
    },
    {
      "field": "uses",
      "target": "DKR8R8Y1RJ"
    },
    {
      "field": "uses",
      "target": "G3XFT0MWKH"
    },
    {
      "field": "uses",
      "target": "REH63MEBNC"
    },
    {
      "field": "uses",
      "target": "SC8HT9T2GA"
    },
    {
      "field": "uses",
      "target": "Y19WM1EMS7"
    },
    {
      "field": "uses",
      "target": "ZAFEF4SAJ7"
    },
    {
      "field": "uses",
      "target": "ZH4V0WHSXT"
    }
  ],
  "body": "\n## Summary\n\nA boundary value problem is a differential equation together with conditions on the solution at the boundary of the domain.\nFor an ordinary differential equation, these are conditions at both ends of an interval.\nIt differs from an [[initial value problem|0FXDW992DS]], where all conditions are given at one point.\nA boundary value problem may have no solution, exactly one, or infinitely many, and the answer depends on the whole interval.\nSuch problems describe equilibria, such as a string fixed at both ends, and lead to eigenvalue problems and Sturm–Liouville theory.\nSee the [[definition|1HGGRS57KT]].\n\n## Explanations\n\n### L1 human\n\nImagine a rope tied to two posts.\nIts shape is decided by conditions at both ends, not by where it starts and in which direction it points.\nFor $y'' + y = 0$ with $y(0) = 0$ and $y(\\frac{\\pi}{2}) = 1$, the only solution is $y = \\sin x$.\nWith $y(0) = 0$ and $y(\\pi) = 1$ instead there is no solution, since every solution with $y(0) = 0$ is $c \\sin x$, which vanishes at $\\pi$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "34B05",
      "34B15"
    ],
    "name": "Boundary value problem",
    "same_as": {
      "wikidata": "Q1332643",
      "wikipedia": "Boundary value problem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Boundary_value_problem"
      }
    ],
    "tag": "TM8VQJWGHZ"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/TM8VQJWGHZ/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/TM8VQJWGHZ.md",
  "msc": [
    "34B05",
    "34B15"
  ],
  "name": "Boundary value problem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TM8/TM8VQJWGHZ.md",
  "status": null,
  "tag": "TM8VQJWGHZ",
  "uri": "m2://TM8VQJWGHZ",
  "url": "https://math2.tamnd.com/wiki/boundary-value-problem/",
  "writing": {
    "tag": "TM8VQJWGHZ",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}