{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0E1ZZBHZ3Z"
    }
  ],
  "body": "\n## Definition\n\nLet $I$ be an open interval, let $P, Q : I \\to \\mathbb{R}$ be continuous functions, and let $n$ be a real number.\nA Bernoulli equation is an [[ordinary differential equation|0P97V20RW7]] of the form\n$$y' + P(x)\\, y = Q(x)\\, y^{n}$$\nfor an unknown function $y$ on a subinterval of $I$. If $n$ is not an integer, the function must satisfy $y \u003e 0$.\n\n## Remarks\n\nFor $n = 0$ and $n = 1$ the equation is [[linear|0Z3YKRQZ48]]. Some authors exclude these two values.\nFor $n \\ne 1$, the substitution $v = y^{1 - n}$ changes the equation, on the set where $y \\ne 0$, into the linear equation $v' + (1 - n) P(x)\\, v = (1 - n) Q(x)$.\nFor $n \u003e 0$, the function $y = 0$ is also a solution.\nThe [[logistic equation|0CC4VC7KR3]] is a Bernoulli equation with $n = 2$.\n\n## Explanations\n\n### L1 human\n\nA Bernoulli equation is a linear equation with an extra power of $y$ on the right side.\nIt is not linear, but a change of the unknown makes it linear.\nThen the [[solution formula|06B9W45XV0]] for linear equations solves it.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0T92VGGVWP",
    "kind": "definition",
    "msc": [
      "34A34"
    ],
    "name": "Bernoulli equation",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Bernoulli_differential_equation"
      }
    ],
    "tag": "0ZB84QCDKR",
    "uses": [
      "0P97V20RW7",
      "0Z3YKRQZ48"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0ZB84QCDKR/",
  "links": [
    {
      "field": "defines",
      "target": "0T92VGGVWP"
    },
    {
      "field": "uses",
      "target": "0P97V20RW7"
    },
    {
      "field": "uses",
      "target": "0Z3YKRQZ48"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0ZB84QCDKR.md",
  "msc": [
    "34A34"
  ],
  "name": "Bernoulli equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0ZB/0ZB84QCDKR.md",
  "status": null,
  "tag": "0ZB84QCDKR",
  "uri": "m2://0ZB84QCDKR",
  "url": "https://math2.tamnd.com/wiki/bernoulli-equation-definition/",
  "writing": {
    "tag": "0ZB84QCDKR",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}