{
  "index": "1966-B-6",
  "type": "ANA",
  "tag": [
    "ANA"
  ],
  "difficulty": "",
  "question": "\\text { B-6. Show that all solutions of the differential equation } y^{\\prime \\prime}+e^{x} y=0 \\text { remain bounded as } x \\rightarrow \\infty \\text {. }",
  "solution": "B-6 Multiplying the equation by \\( e^{-x} y^{\\prime} \\) and integrating from 0 to \\( T \\) we get, after rearranging terms,\n\\[\ny^{2}(T)+2 \\int_{0}^{T} e^{-x} y^{\\prime} y^{\\prime \\prime} d x=y^{2}(0)\n\\]\n\nNote then that for some \\( 0 \\leqq \\zeta \\leqq T \\) we must have\n\\[\n2 \\int_{0}^{T} e^{-x} y^{\\prime} y^{\\prime \\prime} d x=2 \\int_{0}^{\\zeta} y^{\\prime} y^{\\prime \\prime} d x=\\left\\{y^{\\prime}(\\zeta)\\right\\}^{2}-\\left\\{y^{\\prime}(0)\\right\\}^{2}\n\\]\n\nWe then obtain \\( y^{2}(T)+\\left\\{y^{\\prime}(\\zeta)\\right\\}^{2}=y^{2}(0)+\\left\\{y^{\\prime}(0)\\right\\}^{2} \\).",
  "vars": [
    "x",
    "y",
    "\\\\zeta"
  ],
  "params": [
    "T"
  ],
  "sci_consts": [
    "e"
  ],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "abscissa",
        "y": "ordinate",
        "\\zeta": "zetapoint",
        "T": "endpoint"
      },
      "question": "\\text { B-6. Show that all solutions of the differential equation } ordinate^{\\prime \\prime}+e^{abscissa} ordinate=0 \\text { remain bounded as } abscissa \\rightarrow \\infty \\text {. }",
      "solution": "B-6 Multiplying the equation by \\( e^{-abscissa} ordinate^{\\prime} \\) and integrating from 0 to \\( endpoint \\) we get, after rearranging terms,\n\\[\nordinate^{2}(endpoint)+2 \\int_{0}^{endpoint} e^{-abscissa} ordinate^{\\prime} ordinate^{\\prime \\prime} d abscissa=ordinate^{2}(0)\n\\]\n\nNote then that for some \\( 0 \\leqq zetapoint \\leqq endpoint \\) we must have\n\\[\n2 \\int_{0}^{endpoint} e^{-abscissa} ordinate^{\\prime} ordinate^{\\prime \\prime} d abscissa=2 \\int_{0}^{zetapoint} ordinate^{\\prime} ordinate^{\\prime \\prime} d abscissa=\\left\\{ordinate^{\\prime}(zetapoint)\\right\\}^{2}-\\left\\{ordinate^{\\prime}(0)\\right\\}^{2}\n\\]\n\nWe then obtain \\( ordinate^{2}(endpoint)+\\left\\{ordinate^{\\prime}(zetapoint)\\right\\}^{2}=ordinate^{2}(0)+\\left\\{ordinate^{\\prime}(0)\\right\\}^{2} \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "sunflower",
        "y": "raspberry",
        "\\zeta": "raincloud",
        "T": "lighthouse"
      },
      "question": "\\text { B-6. Show that all solutions of the differential equation } raspberry^{\\prime \\prime}+e^{sunflower} raspberry=0 \\text { remain bounded as } sunflower \\rightarrow \\infty \\text {. }",
      "solution": "B-6 Multiplying the equation by \\( e^{-sunflower} raspberry^{\\prime} \\) and integrating from 0 to \\( lighthouse \\) we get, after rearranging terms,\n\\[\nraspberry^{2}(lighthouse)+2 \\int_{0}^{lighthouse} e^{-sunflower} raspberry^{\\prime} raspberry^{\\prime \\prime} d sunflower=raspberry^{2}(0)\n\\]\n\nNote then that for some \\( 0 \\leqq raincloud \\leqq lighthouse \\) we must have\n\\[\n2 \\int_{0}^{lighthouse} e^{-sunflower} raspberry^{\\prime} raspberry^{\\prime \\prime} d sunflower=2 \\int_{0}^{raincloud} raspberry^{\\prime} raspberry^{\\prime \\prime} d sunflower=\\left\\{raspberry^{\\prime}(raincloud)\\right\\}^{2}-\\left\\{raspberry^{\\prime}(0)\\right\\}^{2}\n\\]\n\nWe then obtain \\( raspberry^{2}(lighthouse)+\\left\\{raspberry^{\\prime}(raincloud)\\right\\}^{2}=raspberry^{2}(0)+\\left\\{raspberry^{\\prime}(0)\\right\\}^{2} \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "dependentvar",
        "y": "independentvar",
        "\\zeta": "infinitypoint",
        "T": "starttime"
      },
      "question": "\\text { B-6. Show that all solutions of the differential equation } independentvar^{\\prime \\prime}+e^{dependentvar} independentvar=0 \\text { remain bounded as } dependentvar \\rightarrow \\infty \\text {. }",
      "solution": "B-6 Multiplying the equation by \\( e^{-dependentvar} independentvar^{\\prime} \\) and integrating from 0 to \\( starttime \\) we get, after rearranging terms,\n\\[\nindependentvar^{2}(starttime)+2 \\int_{0}^{starttime} e^{-dependentvar} independentvar^{\\prime} independentvar^{\\prime \\prime} d dependentvar=independentvar^{2}(0)\n\\]\n\nNote then that for some \\( 0 \\leqq infinitypoint \\leqq starttime \\) we must have\n\\[\n2 \\int_{0}^{starttime} e^{-dependentvar} independentvar^{\\prime} independentvar^{\\prime \\prime} d dependentvar=2 \\int_{0}^{infinitypoint} independentvar^{\\prime} independentvar^{\\prime \\prime} d dependentvar=\\left\\{independentvar^{\\prime}(infinitypoint)\\right\\}^{2}-\\left\\{independentvar^{\\prime}(0)\\right\\}^{2}\n\\]\n\nWe then obtain \\( independentvar^{2}(starttime)+\\left\\{independentvar^{\\prime}(infinitypoint)\\right\\}^{2}=independentvar^{2}(0)+\\left\\{independentvar^{\\prime}(0)\\right\\}^{2} \\)."
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "y": "hjgrksla",
        "\\zeta": "mbvcxzas",
        "T": "plokmijn"
      },
      "question": "\\text { B-6. Show that all solutions of the differential equation } hjgrksla^{\\prime \\prime}+e^{qzxwvtnp} hjgrksla=0 \\text { remain bounded as } qzxwvtnp \\rightarrow \\infty \\text {. }",
      "solution": "B-6 Multiplying the equation by \\( e^{-qzxwvtnp} hjgrksla^{\\prime} \\) and integrating from 0 to \\( plokmijn \\) we get, after rearranging terms,\n\\[\nhjgrksla^{2}(plokmijn)+2 \\int_{0}^{plokmijn} e^{-qzxwvtnp} hjgrksla^{\\prime} hjgrksla^{\\prime \\prime} d qzxwvtnp=hjgrksla^{2}(0)\n\\]\n\nNote then that for some \\( 0 \\leqq mbvcxzas \\leqq plokmijn \\) we must have\n\\[\n2 \\int_{0}^{plokmijn} e^{-qzxwvtnp} hjgrksla^{\\prime} hjgrksla^{\\prime \\prime} d qzxwvtnp=2 \\int_{0}^{mbvcxzas} hjgrksla^{\\prime} hjgrksla^{\\prime \\prime} d qzxwvtnp=\\left\\{hjgrksla^{\\prime}(mbvcxzas)\\right\\}^{2}-\\left\\{hjgrksla^{\\prime}(0)\\right\\}^{2}\n\\]\n\nWe then obtain \\( hjgrksla^{2}(plokmijn)+\\left\\{hjgrksla^{\\prime}(mbvcxzas)\\right\\}^{2}=hjgrksla^{2}(0)+\\left\\{hjgrksla^{\\prime}(0)\\right\\}^{2} \\)."
    },
    "kernel_variant": {
      "question": "Let y be a twice–differentiable function on [1,\\infty) satisfying the ordinary differential equation\n\\[\n\\boxed{\\;y''(x)+(1+x^{2})\\,y(x)=0\\,}\\tag{\\*}\n\\]\nShow that y(x) remains bounded as x\\to\\infty; i.e.\n\\[\\sup_{x\\ge 1}|y(x)|<\\infty.\\]",
      "solution": "Correct proof.  Let y satisfy\n\ny″(x)+[1+x²]y(x)=0,    x≥1.\n\nDefine the “energy” function\n\nE(x) = y(x)² + \\frac{y′(x)²}{1+x²}.\n\nCompute its derivative: using y″=−(1+x²)y,\n\nE′(x)\n  = 2y·y′\n    + d/dx(\\frac{y′²}{1+x²})\n  = 2y y′ + \\frac{2y′y″}{1+x²} - \\frac{2x y′²}{(1+x²)²}\n  = 2y y′ - 2y y′ - \\frac{2x y′²}{(1+x²)²}\n  = -\\frac{2x}{(1+x²)²}y′(x)² ≤ 0\n\nfor all x≥1.  Thus E(x) is nonincreasing on [1,∞), so for every x≥1\n\nE(x) ≤ E(1) = y(1)² + \\frac{y′(1)²}{2} < ∞.\n\nIn particular\n\n|y(x)|² ≤ E(x) ≤ y(1)² + \\tfrac12y′(1)²,\n\nso y(x) remains bounded as x→∞, as required.  □",
      "_meta": {
        "core_steps": [
          "Multiply the ODE by an integrating factor (e^{-x})·y′ and integrate over [x0, T].",
          "Rearrange to an identity of the form  y²(T)+2∫ e^{-x} y′y″ dx = y²(x0).",
          "Apply the mean–value (or intermediate-value) argument to replace the weighted integral by an un-weighted one on [x0, ζ] for some ζ∈[x0,T].",
          "Evaluate ∫ y′y″ dx = ½[(y′)²] to obtain y²(T)+(y′(ζ))² = y²(x0)+(y′(x0))².",
          "Conclude |y(T)| is bounded by the constant y²(x0)+(y′(x0))²."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Choice of the left endpoint of integration / point where initial data are taken.",
            "original": "0"
          },
          "slot2": {
            "description": "Specific coefficient e^{x} (with integrating factor e^{-x}); any positive increasing function a(x) whose reciprocal is continuous, positive, and strictly decreasing works identically.",
            "original": "e^{x}"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}