{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0X4NT20S3G"
    },
    {
      "field": "uses",
      "target": "0XRB469Z32"
    },
    {
      "field": "uses",
      "target": "0Y6TH5PQVS"
    }
  ],
  "body": "\n## Statement\n\nLet $I$ be an open interval, let $a_0, \\dots, a_{n-1}, b : I \\to \\mathbb{R}$ be continuous, and let $L y = y^{(n)} + a_{n-1}(x) y^{(n-1)} + \\dots + a_0(x) y$.\nLet $y_p$ be a solution of $L y = b$ on $I$, and let $(y_1, \\dots, y_n)$ be a [[fundamental system|0H4KNNCHPY]] of $L y = 0$ on $I$.\nThen a function $y : I \\to \\mathbb{R}$ is a solution of $L y = b$ on $I$ if and only if\n$$y = y_p + c_1 y_1 + \\dots + c_n y_n$$\nfor some real numbers $c_1, \\dots, c_n$.\n\n## Remarks\n\nProof: $y$ solves $L y = b$ if and only if $L(y - y_p) = 0$, by linearity. Then $y - y_p$ is a combination of the fundamental system.\nThe [[method of undetermined coefficients|0XRB469Z32]] and [[variation of parameters|0Y6TH5PQVS]] find a particular solution $y_p$.\nExample: $y'' + y = 1$ has $y_p = 1$, so its solutions are $1 + c_1 \\cos x + c_2 \\sin x$.\n\n## Explanations\n\n### L1 human\n\nTo solve a linear equation with a right side, find one solution by any method.\nThen add all the solutions of the equation without right side.\nThis gives every solution, and each only one time.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "34A30"
    ],
    "name": "General solution of an inhomogeneous linear equation",
    "role": "theorem",
    "sources": [
      {
        "locator": "Linear differential operator",
        "url": "https://en.wikipedia.org/wiki/Linear_differential_equation"
      }
    ],
    "tag": "0GHWEH6H15",
    "uses": [
      "0PB7K8M7FG",
      "0EQZSAB9EC",
      "0H4KNNCHPY",
      "0F2VBPXNP9"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0GHWEH6H15/",
  "links": [
    {
      "field": "uses",
      "target": "0PB7K8M7FG"
    },
    {
      "field": "uses",
      "target": "0EQZSAB9EC"
    },
    {
      "field": "uses",
      "target": "0H4KNNCHPY"
    },
    {
      "field": "uses",
      "target": "0F2VBPXNP9"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0GHWEH6H15.md",
  "msc": [
    "34A30"
  ],
  "name": "General solution of an inhomogeneous linear equation",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0GH/0GHWEH6H15.md",
  "status": null,
  "tag": "0GHWEH6H15",
  "uri": "m2://0GHWEH6H15",
  "url": "https://math2.tamnd.com/wiki/general-solution-of-an-inhomogeneous-linear-equation/",
  "writing": {
    "tag": "0GHWEH6H15",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}