{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "02SRXDAM3X"
    },
    {
      "field": "uses",
      "target": "08RT9GDDPQ"
    },
    {
      "field": "uses",
      "target": "0F2VBPXNP9"
    },
    {
      "field": "uses",
      "target": "0GHWEH6H15"
    },
    {
      "field": "uses",
      "target": "0GK68PZ6KH"
    },
    {
      "field": "uses",
      "target": "0H4KNNCHPY"
    },
    {
      "field": "uses",
      "target": "0MHE4B1QCQ"
    },
    {
      "field": "uses",
      "target": "0VGWKX2JHB"
    }
  ],
  "body": "\n## Definition\n\nLet $I$ be an open interval and let $a_0, \\dots, a_{n-1} : I \\to \\mathbb{R}$ be continuous functions.\nA homogeneous linear equation of order $n$ is a [[linear differential equation|0PB7K8M7FG]] with right side $0$:\n$$y^{(n)} + a_{n-1}(x)\\, y^{(n-1)} + \\dots + a_1(x)\\, y' + a_0(x)\\, y = 0.$$\nFor a linear equation $L y = b$, the equation $L y = 0$ is the associated homogeneous equation.\n\n## Remarks\n\nThe zero function is always a solution.\nBy the [[superposition principle|0F2VBPXNP9]], the solutions on $I$ form a vector space. It has [[dimension n|0GK68PZ6KH]].\nExample: $y'' - y = 0$ is homogeneous, and its solutions are $C_1 e^{x} + C_2 e^{-x}$.\n\n## Explanations\n\n### L1 human\n\nHomogeneous means that the equation has no term without $y$.\nSo if $y$ is a solution, then $2y$ and $-y$ are solutions, and so is the sum of two solutions.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0KP0QNXPRT",
    "kind": "definition",
    "msc": [
      "34A30"
    ],
    "name": "Homogeneous linear equation",
    "sources": [
      {
        "locator": "Basic terminology",
        "url": "https://en.wikipedia.org/wiki/Linear_differential_equation"
      }
    ],
    "tag": "0EQZSAB9EC",
    "uses": [
      "0PB7K8M7FG"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0EQZSAB9EC/",
  "links": [
    {
      "field": "defines",
      "target": "0KP0QNXPRT"
    },
    {
      "field": "uses",
      "target": "0PB7K8M7FG"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0EQZSAB9EC.md",
  "msc": [
    "34A30"
  ],
  "name": "Homogeneous linear equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0EQ/0EQZSAB9EC.md",
  "status": null,
  "tag": "0EQZSAB9EC",
  "uri": "m2://0EQZSAB9EC",
  "url": "https://math2.tamnd.com/wiki/homogeneous-linear-equation-definition/",
  "writing": {
    "tag": "0EQZSAB9EC",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}