{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0GHWEH6H15"
    },
    {
      "field": "uses",
      "target": "0GK68PZ6KH"
    }
  ],
  "body": "\n## Statement\n\nLet $I$ be an open interval, and let $L y = y^{(n)} + a_{n-1}(x) y^{(n-1)} + \\dots + a_0(x) y$ with continuous functions $a_0, \\dots, a_{n-1}$ on $I$.\nLet $y_1$ and $y_2$ be solutions of the [[homogeneous linear equation|0EQZSAB9EC]] $L y = 0$ on $I$, and let $c_1, c_2$ be real numbers. Then $c_1 y_1 + c_2 y_2$ is a solution of $L y = 0$ on $I$.\n\n## Remarks\n\nThe proof is the linearity of $L$: $L(c_1 y_1 + c_2 y_2) = c_1 L y_1 + c_2 L y_2 = 0$.\nThe same argument gives a variant for right sides: if $L y_1 = b_1$ and $L y_2 = b_2$, then $L(y_1 + y_2) = b_1 + b_2$.\nThe principle fails for nonlinear equations. For $y' = y^2$, the function $y = -1/x$ is a solution on $x \u003e 0$, but $2y$ is not.\n\n## Explanations\n\n### L1 human\n\nFor a linear equation, you can add solutions and scale them, and you get solutions again.\nIn physics this is superposition: two waves that solve a linear wave equation add up to a third solution.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "kind": "statement",
    "msc": [
      "34A30"
    ],
    "name": "Superposition principle",
    "role": "proposition",
    "same_as": {
      "wikidata": "Q284885",
      "wikipedia": "Superposition principle"
    },
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Superposition_principle"
      }
    ],
    "tag": "0F2VBPXNP9",
    "uses": [
      "0EQZSAB9EC",
      "0PB7K8M7FG"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0F2VBPXNP9/",
  "links": [
    {
      "field": "uses",
      "target": "0EQZSAB9EC"
    },
    {
      "field": "uses",
      "target": "0PB7K8M7FG"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0F2VBPXNP9.md",
  "msc": [
    "34A30"
  ],
  "name": "Superposition principle",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0F2/0F2VBPXNP9.md",
  "status": null,
  "tag": "0F2VBPXNP9",
  "uri": "m2://0F2VBPXNP9",
  "url": "https://math2.tamnd.com/wiki/superposition-principle/",
  "writing": {
    "tag": "0F2VBPXNP9",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}