{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $J$ be an open interval and let $F : J \\to \\mathbb{R}$ be a continuous function.\nA homogeneous first-order equation is an [[ordinary differential equation|0P97V20RW7]] of the form\n$$y' = F\\left(\\frac{y}{x}\\right)$$\nfor an unknown function $y$ of $x$. Its domain is an interval that does not contain $0$, with $y(x)/x \\in J$.\n\n## Remarks\n\nAn equivalent form is $M(x, y)\\, dx + N(x, y)\\, dy = 0$, where $M$ and $N$ are homogeneous functions of the same degree $d$: $M(t x, t y) = t^d M(x, y)$ and $N(t x, t y) = t^d N(x, y)$ for $t \\ne 0$.\nThe substitution $y = x v$ gives $x v' = F(v) - v$. This is a [[separable equation|0V4E623THY]] for $v$.\nExample: $y' = (x + y)/x = 1 + y/x$ gives $x v' = 1$, so $v = \\ln|x| + C$ and $y = x \\ln|x| + C x$.\nThis use of the word homogeneous is different from its use for [[homogeneous linear equations|0EQZSAB9EC]].\n\n## Explanations\n\n### L1 human\n\nIn a homogeneous equation, the slope $y'$ depends only on the ratio $y/x$.\nSo the slope field is the same along each line through the origin.\nThe new unknown $v = y/x$ captures this, and the equation becomes separable.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "defines": "0RN88KZBJ3",
    "kind": "definition",
    "msc": [
      "34A05"
    ],
    "name": "Homogeneous first-order equation",
    "sources": [
      {
        "locator": "Homogeneous first-order differential equations",
        "url": "https://en.wikipedia.org/wiki/Homogeneous_differential_equation"
      }
    ],
    "tag": "0K4ZWTFSV5",
    "uses": [
      "0P97V20RW7",
      "0V4E623THY"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0K4ZWTFSV5/",
  "links": [
    {
      "field": "defines",
      "target": "0RN88KZBJ3"
    },
    {
      "field": "uses",
      "target": "0P97V20RW7"
    },
    {
      "field": "uses",
      "target": "0V4E623THY"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0K4ZWTFSV5.md",
  "msc": [
    "34A05"
  ],
  "name": "Homogeneous first-order equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0K4/0K4ZWTFSV5.md",
  "status": null,
  "tag": "0K4ZWTFSV5",
  "uri": "m2://0K4ZWTFSV5",
  "url": "https://math2.tamnd.com/wiki/homogeneous-first-order-equation-definition/",
  "writing": {
    "tag": "0K4ZWTFSV5",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}