{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-ordinary-differential-equation",
  "semantic_family_id": "sf-ordinary-differential-equation",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "常微分方程"
  },
  "summary": {
    "zh-Hant": "常微分方程關聯未知單變數函數及其有限階導數；候選函數的殘差核對不等於存在唯一性定理。"
  },
  "statement": {
    "zh-Hant": "常微分方程是在指定區間上，關聯未知單變數函數 y 與其有限階導數的方程。方程中出現的最高導數階數稱為階數；把候選 y 代入並得零殘差，只核對該候選在指定範圍是否滿足方程。"
  },
  "formula": {
    "tex": "F(y,D(y))=0",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>F</mi><mo>(</mo><mi>y</mi><mo>,</mo><mrow><mi>D</mi><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "F",
          "glyph": "F"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "y",
            "glyph": "y"
          },
          {
            "type": "function_call",
            "callee": {
              "type": "symbol",
              "name": "D",
              "glyph": "D"
            },
            "arguments": [
              {
                "type": "symbol",
                "name": "y",
                "glyph": "y"
              }
            ]
          }
        ]
      },
      "rhs": {
        "type": "number",
        "value": 0
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "ed7078ea8eb520291ed118b83356ff2d7bb6e7c263ea45906ab474951ac644f3"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "未知量為單一實變數的實值函數，題目明示所用導數與係數。"
    },
    {
      "id": "a2",
      "zh-Hant": "題庫限整係數多項式候選與指定點的精確殘差；不由方程本身推論解存在、唯一或全域延拓。"
    }
  ],
  "symbols": [
    {
      "token": "F",
      "role": "differential_relation",
      "role_zh": "微分關係",
      "scope": "representative_formula"
    },
    {
      "token": "y",
      "role": "unknown_function",
      "role_zh": "未知函數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-derivative",
      "title_zh": "實函數在一點的導數與差商",
      "reason_zh": "微分方程以導數表示變化。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "未知量是定義域上的函數。"
    },
    {
      "id": "mko-equation",
      "title_zh": "方程式",
      "reason_zh": "解需使微分關係成立。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "ODE 的階數由實際出現的最高導數決定，不由多項式候選的次數或方程項數決定。",
      "候選解核對應重算 y、y'、y'' 等並代入同一方程；只在一點殘差為零通常不足以證整個區間上為解。",
      "初值可以篩選候選解，但存在與唯一性仍需連續性、Lipschitz 等額外假設及對應定理。"
    ],
    "common_misconception_zh": "把一點代入成功當成區間上的解，或從 ODE 與初值直接宣稱必有唯一解。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪只做多項式候選的階數、指定點殘差與初值核對；不證存在唯一性。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "建立初值問題、解區間、存在唯一條件及標準一階方程類型。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-ordinary-differential-equation.md 整理；2026-10-13 脈衝 R02（使用者手動接續 2026-10-20 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}