{
  "aliases": [
    "euler-cauchy-equation",
    "equidimensional-equation"
  ],
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $a_0, \\dots, a_n$ be real numbers with $a_n \\ne 0$.\nThe Cauchy–Euler equation with these coefficients is the [[linear differential equation|0PB7K8M7FG]]\n$$a_n x^n y^{(n)} + a_{n-1} x^{n-1} y^{(n-1)} + \\dots + a_1 x\\, y' + a_0\\, y = 0$$\nfor an unknown function $y$ on the interval $x \u003e 0$.\n\n## Remarks\n\nThe substitution $x = e^{t}$, $u(t) = y(e^t)$ changes it into a [[linear equation with constant coefficients|0A6PGTTP2A]] for $u$.\nFor $n = 2$, the trial solution $y = x^m$ gives the indicial equation $a_2 m (m - 1) + a_1 m + a_0 = 0$.\nFor order $2$: a double root $m$ gives the solutions $x^m$ and $x^m \\ln x$. Roots $\\alpha \\pm i \\beta$ with $\\beta \\ne 0$ give $x^\\alpha \\cos(\\beta \\ln x)$ and $x^\\alpha \\sin(\\beta \\ln x)$.\nExample: $x^2 y'' - 2 y = 0$ gives $m(m-1) - 2 = 0$, so $m = 2$ or $m = -1$, and the solutions are $c_1 x^2 + c_2 x^{-1}$.\n\n## Explanations\n\n### L1 human\n\nIn this equation, each derivative $y^{(k)}$ is multiplied by $x^k$.\nSo powers $x^m$ behave like exponentials do for constant coefficients: the equation becomes a polynomial equation in $m$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "euler-cauchy-equation",
      "equidimensional-equation"
    ],
    "defines": "03F7N3915D",
    "kind": "definition",
    "msc": [
      "34A30"
    ],
    "name": "Cauchy–Euler equation",
    "sources": [
      {
        "locator": "The equation",
        "url": "https://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation"
      }
    ],
    "tag": "0WDNRBXX7K",
    "uses": [
      "0PB7K8M7FG",
      "0A6PGTTP2A"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0WDNRBXX7K/",
  "links": [
    {
      "field": "defines",
      "target": "03F7N3915D"
    },
    {
      "field": "uses",
      "target": "0PB7K8M7FG"
    },
    {
      "field": "uses",
      "target": "0A6PGTTP2A"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0WDNRBXX7K.md",
  "msc": [
    "34A30"
  ],
  "name": "Cauchy–Euler equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0WD/0WDNRBXX7K.md",
  "status": null,
  "tag": "0WDNRBXX7K",
  "uri": "m2://0WDNRBXX7K",
  "url": "https://math2.tamnd.com/wiki/cauchy-euler-equation-definition/",
  "writing": {
    "tag": "0WDNRBXX7K",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}