{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0BEZX8F2XW"
    },
    {
      "field": "uses",
      "target": "0RNXVVTJTJ"
    }
  ],
  "body": "\n## Statement\n\nLet $a_0, \\dots, a_{n-1}$ be real numbers, and let $p(r) = r^n + a_{n-1} r^{n-1} + \\dots + a_0$ be the [[characteristic polynomial|0QDXMG46QW]] of the equation $y^{(n)} + a_{n-1} y^{(n-1)} + \\dots + a_0 y = 0$.\nLet $r$ be a real root of $p$ of multiplicity $m \\ge 1$.\nThen the $m$ functions $e^{r x}, x e^{r x}, \\dots, x^{m-1} e^{r x}$ are linearly independent solutions of the equation on $\\mathbb{R}$.\nIf all the roots of $p$ are real, the union of these families over the distinct roots of $p$ is a [[fundamental system|0H4KNNCHPY]].\n\n## Remarks\n\nFor order $2$: if $r$ is a double root of $r^2 + a_1 r + a_0$, the solutions are $(c_1 + c_2 x) e^{r x}$.\nExample: $y'' - 2 y' + y = 0$ has $p(r) = (r - 1)^2$, so its solutions are $(c_1 + c_2 x) e^{x}$.\nThe same result holds for complex roots, with complex solutions. For the real form, see [[complex conjugate roots|0BEZX8F2XW]].\n\n## Explanations\n\n### L1 human\n\nA double root gives only one exponential, but we need two solutions.\nThe extra solution is the exponential times $x$.\nA root of multiplicity $m$ gives the exponential times $1, x, \\dots, x^{m-1}$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "34A30"
    ],
    "name": "Solutions for repeated roots",
    "role": "theorem",
    "sources": [
      {
        "locator": "Repeated real roots",
        "url": "https://en.wikipedia.org/wiki/Characteristic_equation_(calculus)"
      }
    ],
    "tag": "0HRVY67CNV",
    "uses": [
      "0QDXMG46QW",
      "0H4KNNCHPY",
      "0A6PGTTP2A",
      "0HJ0HDA6W7"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0HRVY67CNV/",
  "links": [
    {
      "field": "uses",
      "target": "0QDXMG46QW"
    },
    {
      "field": "uses",
      "target": "0H4KNNCHPY"
    },
    {
      "field": "uses",
      "target": "0A6PGTTP2A"
    },
    {
      "field": "uses",
      "target": "0HJ0HDA6W7"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0HRVY67CNV.md",
  "msc": [
    "34A30"
  ],
  "name": "Solutions for repeated roots",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0HR/0HRVY67CNV.md",
  "status": null,
  "tag": "0HRVY67CNV",
  "uri": "m2://0HRVY67CNV",
  "url": "https://math2.tamnd.com/wiki/solutions-for-repeated-roots/",
  "writing": {
    "tag": "0HRVY67CNV",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}