{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\n1. Let $u, v : I \\to \\R$ be $n$ times differentiable and $\\alpha, \\beta \\in \\R$. By [[linearity of differentiation|0M0TKFRX2A]], applied $k$ times, $(\\alpha u + \\beta v)^{(k)} = \\alpha u^{(k)} + \\beta v^{(k)}$ for $k \\le n$.\n2. Multiplying by $a_k$ and adding over $k$ gives $L(\\alpha u + \\beta v) = \\alpha L u + \\beta L v$ on $I$. This is the computation of the [[superposition principle|0F2VBPXNP9]].\n3. Let $S$ be the set of solutions of the [[homogeneous equation|0EQZSAB9EC]] $L y = 0$ on $I$. Since $(y_1, \\dots, y_n)$ is a [[fundamental system|0H4KNNCHPY]], it is a basis of $S$.\n4. Assume $y$ is a solution of the [[linear equation|0PB7K8M7FG]] $L y = b$ on $I$. Then $y - y_p$ is $n$ times differentiable, and $L(y - y_p) = L y - L y_p = b - b = 0$ by step 2.\n5. So $y - y_p \\in S$. By step 3, $y - y_p = c_1 y_1 + \\dots + c_n y_n$ for some real numbers $c_1, \\dots, c_n$. This gives the stated form of $y$.\n6. Conversely, let $y = y_p + c_1 y_1 + \\dots + c_n y_n$ with $c_1, \\dots, c_n \\in \\R$. Then $y$ is $n$ times differentiable, as a combination of such functions.\n7. By step 2, $L y = L y_p + c_1 L y_1 + \\dots + c_n L y_n = b + 0 + \\dots + 0 = b$ on $I$. So $y$ solves $L y = b$ on $I$.\n\n## Idea\n\nThe operator $L$ is linear. So two solutions of $L y = b$ differ by a solution of $L y = 0$. The fundamental system describes all solutions of $L y = 0$, and adding $y_p$ describes all solutions of $L y = b$.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "34A30"
    ],
    "name": "Proof of the general solution of an inhomogeneous linear equation",
    "proves": "0GHWEH6H15",
    "sources": [
      {
        "locator": "Linear differential operator",
        "url": "https://en.wikipedia.org/wiki/Linear_differential_equation"
      }
    ],
    "tag": "FWSXQTTPE6",
    "uses": [
      "0PB7K8M7FG",
      "0EQZSAB9EC",
      "0H4KNNCHPY",
      "0F2VBPXNP9",
      "0M0TKFRX2A"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/FWSXQTTPE6/",
  "links": [
    {
      "field": "proves",
      "target": "0GHWEH6H15"
    },
    {
      "field": "uses",
      "target": "0PB7K8M7FG"
    },
    {
      "field": "uses",
      "target": "0EQZSAB9EC"
    },
    {
      "field": "uses",
      "target": "0H4KNNCHPY"
    },
    {
      "field": "uses",
      "target": "0F2VBPXNP9"
    },
    {
      "field": "uses",
      "target": "0M0TKFRX2A"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/FWSXQTTPE6.md",
  "msc": [
    "34A30"
  ],
  "name": "Proof of the general solution of an inhomogeneous linear equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FWS/FWSXQTTPE6.md",
  "status": null,
  "tag": "FWSXQTTPE6",
  "uri": "m2://FWSXQTTPE6",
  "url": "https://math2.tamnd.com/wiki/proof-of-the-general-solution-of-an-inhomogeneous-linear-equation/",
  "writing": {
    "tag": "FWSXQTTPE6",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}