{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0E4GXX7SJP"
    },
    {
      "field": "uses",
      "target": "0K4ZWTFSV5"
    }
  ],
  "body": "\n## Definition\n\nLet $I$ and $J$ be open [[intervals|0Q1D85B8G6]] of $\\mathbb{R}$, and let $g : I \\to \\mathbb{R}$ and $h : J \\to \\mathbb{R}$ be continuous functions.\nA separable equation is a first-order [[ordinary differential equation|0P97V20RW7]] of the form\n$$y' = g(x)\\, h(y),$$\nfor an unknown function $y$ of $x$ with values in $J$.\n\n## Remarks\n\nMethod: if $h(y) \\ne 0$, write the equation as $y'/h(y) = g(x)$ and integrate. With antiderivatives $H$ of $1/h$ and $G$ of $g$, the solutions satisfy $H(y(x)) = G(x) + C$ for a constant $C$.\nFor each zero $c$ of $h$, the constant function $y = c$ is a solution. The method above does not find these solutions.\nExample: $y' = x y$ has the solutions $y = C e^{x^2/2}$ with a real constant $C$.\n[[Exponential growth|00SVP6W3CB]] and the [[logistic equation|0CC4VC7KR3]] are separable.\n\n## Explanations\n\n### L1 human\n\nIn a separable equation, the right side is a product: one factor depends only on $x$ and the other only on $y$.\nPut everything with $y$ on one side and everything with $x$ on the other side, then integrate both sides.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "074E9A8ZJK",
    "kind": "definition",
    "msc": [
      "34A05"
    ],
    "name": "Separable equation",
    "sources": [
      {
        "locator": "Ordinary differential equations (ODE)",
        "url": "https://en.wikipedia.org/wiki/Separation_of_variables"
      }
    ],
    "tag": "0V4E623THY",
    "uses": [
      "0P97V20RW7",
      "0Q1D85B8G6"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0V4E623THY/",
  "links": [
    {
      "field": "defines",
      "target": "074E9A8ZJK"
    },
    {
      "field": "uses",
      "target": "0P97V20RW7"
    },
    {
      "field": "uses",
      "target": "0Q1D85B8G6"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0V4E623THY.md",
  "msc": [
    "34A05"
  ],
  "name": "Separable equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0V4/0V4E623THY.md",
  "status": null,
  "tag": "0V4E623THY",
  "uri": "m2://0V4E623THY",
  "url": "https://math2.tamnd.com/wiki/separable-equation-definition/",
  "writing": {
    "tag": "0V4E623THY",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}