{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Summary\n\nReduction of order is a method to find a second solution of a homogeneous linear equation from a known one.\nLet $p, q$ be continuous on an open interval $I$, and let $y_1$ be a solution of $y'' + p(x) y' + q(x) y = 0$ with no zero on $I$.\nPut $y = v\\, y_1$. Then $y$ is a solution if and only if $y_1 v'' + (2 y_1' + p\\, y_1) v' = 0$, a first-order linear equation for $v'$.\nOne solution gives $y_2 = y_1 \\int \\frac{e^{-\\int p\\, dx}}{y_1^2}\\, dx$, and $(y_1, y_2)$ is a [[fundamental system|0H4KNNCHPY]] on $I$.\n\n## Remarks\n\nExample: $y_1 = e^{x}$ solves $y'' - 2 y' + y = 0$. Here $p = -2$, so $v' = e^{2x}/e^{2x} = 1$, and $y_2 = x e^x$.\nThe [[Wronskian|0N5NF1RGBP]] of $y_1$ and $y_2$ is $e^{-\\int p\\, dx}$, which has no zero. This is consistent with [[Abel's identity|0VGWKX2JHB]].\nThe same idea lowers the order by one for a linear equation of order $n$.\n\n## Explanations\n\n### L1 human\n\nIf you know one solution, look for a second one as the first one times an unknown function $v$.\nThe terms without derivatives of $v$ cancel, because $y_1$ is a solution.\nWhat remains is a first-order equation for $v'$, which is easy to solve.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "concept",
    "msc": [
      "34A30"
    ],
    "name": "Reduction of order",
    "same_as": {
      "wikidata": "Q1430365",
      "wikipedia": "Reduction of order"
    },
    "sources": [
      {
        "locator": "Second-order linear ordinary differential equations",
        "url": "https://en.wikipedia.org/wiki/Reduction_of_order"
      }
    ],
    "tag": "02SRXDAM3X",
    "uses": [
      "0EQZSAB9EC",
      "0H4KNNCHPY",
      "0N5NF1RGBP"
    ]
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/02SRXDAM3X/",
  "links": [
    {
      "field": "uses",
      "target": "0EQZSAB9EC"
    },
    {
      "field": "uses",
      "target": "0H4KNNCHPY"
    },
    {
      "field": "uses",
      "target": "0N5NF1RGBP"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/02SRXDAM3X.md",
  "msc": [
    "34A30"
  ],
  "name": "Reduction of order",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/02S/02SRXDAM3X.md",
  "status": null,
  "tag": "02SRXDAM3X",
  "uri": "m2://02SRXDAM3X",
  "url": "https://math2.tamnd.com/wiki/reduction-of-order/",
  "writing": {
    "tag": "02SRXDAM3X",
    "kind": "concept",
    "level": "W1",
    "next": [
      {
        "what": "edge defines in",
        "note": "at least one definition",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of in",
        "note": "at least two examples",
        "have": 0,
        "want": 2
      },
      {
        "what": "edge not_instance_of in",
        "note": "at least one non-example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge specializes both",
        "note": "the more general notions and the special cases",
        "have": 0,
        "want": 1
      }
    ]
  }
}