{
  "index": "1982-A-4",
  "type": "ANA",
  "tag": [
    "ANA"
  ],
  "difficulty": "",
  "question": "Problem A-4\nAssume that the system of simultaneous differential equations\n\\[\ny^{\\prime}=-z^{3}, \\quad z^{\\prime}=y^{3}\n\\]\nwith the initial conditions \\( y(0)=1, z(0)=0 \\) has a unique solution \\( y=f(x), z=g(x) \\) defined for all real \\( x \\). Prove that there exists a positive constant \\( L \\) such that for all real \\( x \\).\n\\[\nf(x+L)=f(x), \\quad g(x+L)=g(x)\n\\]",
  "solution": "A-4.\nThe differential equations imply that\n\\[\ny^{3} y^{\\prime}+z^{3} z^{\\prime}=z^{\\prime} y^{\\prime}-y^{\\prime} z^{\\prime}=0\n\\]\nand hence that \\( y^{4}+z^{4} \\) is constant. This and the initial conditions give \\( y^{4}+z^{4}=1 \\). Thinking of \\( x \\) as a time variable and \\( (y, z) \\) as the coordinates of a point in a plane, this point moves on the curve \\( y^{4}+z^{4}=1 \\) with speed\n\\[\n\\left[\\left(y^{\\prime}\\right)^{2}+\\left(z^{\\prime}\\right)^{2}\\right]^{1 / 2}=\\left(z^{6}+y^{6}\\right)^{1 / 2} .\n\\]\n\nAt any time, either \\( y^{4} \\) or \\( z^{4} \\) is at least \\( \\frac{1}{2} \\) and so the speed is at least \\( \\left\\{\\left[\\left(\\frac{1}{2}\\right)^{1 / 4}\\right]^{6}\\right\\}^{1 / 2} \\). Hence the point will go completely around the finite curve in some time \\( L \\). As the speed depends only on \\( y \\) and \\( z \\) (and not on \\( x \\) ), the motion is periodic with period \\( L \\).",
  "vars": [
    "x",
    "y",
    "z",
    "f",
    "g"
  ],
  "params": [
    "L"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "realaxis",
        "y": "firstcoord",
        "z": "secondcoord",
        "f": "firstmap",
        "g": "secondmap",
        "L": "periodlen"
      },
      "question": "Problem A-4\nAssume that the system of simultaneous differential equations\n\\[\nfirstcoord^{\\prime}=-secondcoord^{3}, \\quad secondcoord^{\\prime}=firstcoord^{3}\n\\]\nwith the initial conditions \\( firstcoord(0)=1, secondcoord(0)=0 \\) has a unique solution \\( firstcoord=firstmap(realaxis), secondcoord=secondmap(realaxis) \\) defined for all real realaxis. Prove that there exists a positive constant periodlen such that for all real realaxis.\n\\[\nfirstmap(realaxis+periodlen)=firstmap(realaxis), \\quad secondmap(realaxis+periodlen)=secondmap(realaxis)\n\\]",
      "solution": "A-4.\nThe differential equations imply that\n\\[\nfirstcoord^{3} firstcoord^{\\prime}+secondcoord^{3} secondcoord^{\\prime}=secondcoord^{\\prime} firstcoord^{\\prime}-firstcoord^{\\prime} secondcoord^{\\prime}=0\n\\]\nand hence that \\( firstcoord^{4}+secondcoord^{4} \\) is constant. This and the initial conditions give \\( firstcoord^{4}+secondcoord^{4}=1 \\). Thinking of realaxis as a time variable and \\( (firstcoord, secondcoord) \\) as the coordinates of a point in a plane, this point moves on the curve \\( firstcoord^{4}+secondcoord^{4}=1 \\) with speed\n\\[\n\\left[\\left(firstcoord^{\\prime}\\right)^{2}+\\left(secondcoord^{\\prime}\\right)^{2}\\right]^{1 / 2}=\\left(secondcoord^{6}+firstcoord^{6}\\right)^{1 / 2} .\n\\]\n\nAt any time, either \\( firstcoord^{4} \\) or \\( secondcoord^{4} \\) is at least \\( \\frac{1}{2} \\) and so the speed is at least \\( \\left\\{\\left[\\left(\\frac{1}{2}\\right)^{1 / 4}\\right]^{6}\\right\\}^{1 / 2} \\). Hence the point will go completely around the finite curve in some time periodlen. As the speed depends only on firstcoord and secondcoord (and not on realaxis), the motion is periodic with period periodlen."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "longitude",
        "y": "crimsonleaf",
        "z": "turquoise",
        "f": "ampersand",
        "g": "whirlwind",
        "L": "dandelion"
      },
      "question": "Problem A-4\nAssume that the system of simultaneous differential equations\n\\[\ncrimsonleaf^{\\prime}=-turquoise^{3}, \\quad turquoise^{\\prime}=crimsonleaf^{3}\n\\]\nwith the initial conditions \\( crimsonleaf(0)=1, turquoise(0)=0 \\) has a unique solution \\( crimsonleaf=ampersand(longitude), turquoise=whirlwind(longitude) \\) defined for all real \\( longitude \\). Prove that there exists a positive constant \\( dandelion \\) such that for all real \\( longitude \\).\n\\[\nampersand(longitude+dandelion)=ampersand(longitude), \\quad whirlwind(longitude+dandelion)=whirlwind(longitude)\n\\]",
      "solution": "A-4.\nThe differential equations imply that\n\\[\ncrimsonleaf^{3} crimsonleaf^{\\prime}+turquoise^{3} turquoise^{\\prime}=turquoise^{\\prime} crimsonleaf^{\\prime}-crimsonleaf^{\\prime} turquoise^{\\prime}=0\n\\]\nand hence that \\( crimsonleaf^{4}+turquoise^{4} \\) is constant. This and the initial conditions give \\( crimsonleaf^{4}+turquoise^{4}=1 \\). Thinking of \\( longitude \\) as a time variable and \\( (crimsonleaf, turquoise) \\) as the coordinates of a point in a plane, this point moves on the curve \\( crimsonleaf^{4}+turquoise^{4}=1 \\) with speed\n\\[\n\\left[\\left(crimsonleaf^{\\prime}\\right)^{2}+\\left(turquoise^{\\prime}\\right)^{2}\\right]^{1 / 2}=\\left(turquoise^{6}+crimsonleaf^{6}\\right)^{1 / 2} .\n\\]\n\nAt any time, either \\( crimsonleaf^{4} \\) or \\( turquoise^{4} \\) is at least \\( \\frac{1}{2} \\) and so the speed is at least \\( \\left\\{\\left[\\left(\\frac{1}{2}\\right)^{1 / 4}\\right]^{6}\\right\\}^{1 / 2} \\). Hence the point will go completely around the finite curve in some time \\( dandelion \\). As the speed depends only on \\( crimsonleaf \\) and \\( turquoise \\) (and not on \\( longitude \\) ), the motion is periodic with period \\( dandelion \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "fixedpoint",
        "y": "horizontalaxis",
        "z": "flatplane",
        "f": "constantvalue",
        "g": "steadyform",
        "L": "negativebound"
      },
      "question": "Problem A-4\nAssume that the system of simultaneous differential equations\n\\[\nhorizontalaxis^{\\prime}=-flatplane^{3}, \\quad flatplane^{\\prime}=horizontalaxis^{3}\n\\]\nwith the initial conditions \\( horizontalaxis(0)=1, flatplane(0)=0 \\) has a unique solution \\( horizontalaxis=constantvalue(fixedpoint), flatplane=steadyform(fixedpoint) \\) defined for all real \\( fixedpoint \\). Prove that there exists a positive constant \\( negativebound \\) such that for all real \\( fixedpoint \\).\n\\[\nconstantvalue(fixedpoint+negativebound)=constantvalue(fixedpoint), \\quad steadyform(fixedpoint+negativebound)=steadyform(fixedpoint)\n\\]",
      "solution": "A-4.\nThe differential equations imply that\n\\[\nhorizontalaxis^{3} horizontalaxis^{\\prime}+flatplane^{3} flatplane^{\\prime}=flatplane^{\\prime} horizontalaxis^{\\prime}-horizontalaxis^{\\prime} flatplane^{\\prime}=0\n\\]\nand hence that \\( horizontalaxis^{4}+flatplane^{4} \\) is constant. This and the initial conditions give \\( horizontalaxis^{4}+flatplane^{4}=1 \\). Thinking of \\( fixedpoint \\) as a time variable and \\( (horizontalaxis, flatplane) \\) as the coordinates of a point in a plane, this point moves on the curve \\( horizontalaxis^{4}+flatplane^{4}=1 \\) with speed\n\\[\n\\left[\\left(horizontalaxis^{\\prime}\\right)^{2}+\\left(flatplane^{\\prime}\\right)^{2}\\right]^{1 / 2}=\\left(flatplane^{6}+horizontalaxis^{6}\\right)^{1 / 2} .\n\\]\n\nAt any time, either \\( horizontalaxis^{4} \\) or \\( flatplane^{4} \\) is at least \\( \\frac{1}{2} \\) and so the speed is at least \\( \\left\\{\\left[\\left(\\frac{1}{2}\\right)^{1 / 4}\\right]^{6}\\right\\}^{1 / 2} \\). Hence the point will go completely around the finite curve in some time \\( negativebound \\). As the speed depends only on \\( horizontalaxis \\) and \\( flatplane \\) (and not on \\( fixedpoint \\) ), the motion is periodic with period \\( negativebound \\)."
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "y": "hjgrksla",
        "z": "mvndtjqe",
        "f": "rclzsmep",
        "g": "wptnjkdu",
        "L": "bxmragfe"
      },
      "question": "Problem A-4\nAssume that the system of simultaneous differential equations\n\\[\nhjgrksla^{\\prime}=-mvndtjqe^{3}, \\quad mvndtjqe^{\\prime}=hjgrksla^{3}\n\\]\nwith the initial conditions \\( hjgrksla(0)=1, mvndtjqe(0)=0 \\) has a unique solution \\( hjgrksla=rclzsmep(qzxwvtnp), mvndtjqe=wptnjkdu(qzxwvtnp) \\) defined for all real \\( qzxwvtnp \\). Prove that there exists a positive constant \\( bxmragfe \\) such that for all real \\( qzxwvtnp \\).\n\\[\nrclzsmep(qzxwvtnp+bxmragfe)=rclzsmep(qzxwvtnp), \\quad wptnjkdu(qzxwvtnp+bxmragfe)=wptnjkdu(qzxwvtnp)\n\\]",
      "solution": "A-4.\nThe differential equations imply that\n\\[\nhjgrksla^{3} hjgrksla^{\\prime}+mvndtjqe^{3} mvndtjqe^{\\prime}=mvndtjqe^{\\prime} hjgrksla^{\\prime}-hjgrksla^{\\prime} mvndtjqe^{\\prime}=0\n\\]\nand hence that \\( hjgrksla^{4}+mvndtjqe^{4} \\) is constant. This and the initial conditions give \\( hjgrksla^{4}+mvndtjqe^{4}=1 \\). Thinking of \\( qzxwvtnp \\) as a time variable and \\( (hjgrksla, mvndtjqe) \\) as the coordinates of a point in a plane, this point moves on the curve \\( hjgrksla^{4}+mvndtjqe^{4}=1 \\) with speed\n\\[\n\\left[\\left(hjgrksla^{\\prime}\\right)^{2}+\\left(mvndtjqe^{\\prime}\\right)^{2}\\right]^{1 / 2}=\\left(mvndtjqe^{6}+hjgrksla^{6}\\right)^{1 / 2} .\n\\]\n\nAt any time, either \\( hjgrksla^{4} \\) or \\( mvndtjqe^{4} \\) is at least \\( \\frac{1}{2} \\) and so the speed is at least \\( \\left\\{\\left[\\left(\\frac{1}{2}\\right)^{1 / 4}\\right]^{6}\\right\\}^{1 / 2} \\). Hence the point will go completely around the finite curve in some time \\( bxmragfe \\). As the speed depends only on hjgrksla and mvndtjqe (and not on qzxwvtnp), the motion is periodic with period \\( bxmragfe \\)."
    },
    "kernel_variant": {
      "question": "Let \\(f,g:\\mathbb R\\to\\mathbb R\\) be the unique solution of the system of differential equations\n\\[\n\\boxed{\\;y'=f'(x)=-z^{5},\\qquad z'=g'(x)=y^{5}\\;}\\tag{1}\n\\]\nwith the initial conditions\n\\[\nf(0)=1,\\qquad g(0)=1.\n\\]\nProve that there is a positive constant \\(L\\) such that for every real \\(x\\)\n\\[\nf(x+L)=f(x),\\qquad g(x+L)=g(x).\\]",
      "solution": "1. First integral.\n   Differentiate y^6+z^6:\n   d/dx(y^6+z^6)=6(y^5y'+z^5z')=6(y^5(−z^5)+z^5y^5)=0.\n   Hence y^6+z^6 is constant.  The initial data give y^6(0)+z^6(0)=1+1=2, so\n   y^6+z^6=2 for all x.\n\n2. Closed trajectory.\n   Thus (y(x),z(x)) moves on the compact smooth curve C={ (u,v): u^6+v^6=2 }.\n\n3. Speed depends only on position.\n   From y'=−z^5, z'=y^5 we get\n   speed v(x)=√((y')^2+(z')^2)=√(z^10+y^10),\n   a function of (y,z) alone.\n\n4. Strictly positive lower bound.\n   Since y^6+z^6=2, at least one of |y|,|z|≥1, so y^10+z^10≥1 and v(x)≥1.\n\n5. Finite traversal time.\n   The curve C has finite length ℓ.  Traveling at speed≥1, the trajectory must return to (1,1) in time L≤ℓ.\n\n6. Periodicity.\n   The system is autonomous.  Once (y,z) returns to its initial state, uniqueness forces the motion to repeat.  Hence f(x+L)=f(x), g(x+L)=g(x) for all x, as desired.",
      "_meta": {
        "core_steps": [
          "Compute (y^{k+1} + z^{k+1})' = 0 ⇒ motion stays on closed level curve",
          "Use initial data to identify the particular curve y^{k+1}+z^{k+1}=C",
          "Write speed v(x)=√(y'²+z'²)=√(y^{2k}+z^{2k}), a function of position only",
          "Since y^{k+1}+z^{k+1}=C, at least one term ≥ C/2 ⇒ v(x) ≥ positive bound",
          "Finite curve length / positive speed ⇒ loop completed in finite L; thus (y,z) is L-periodic"
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Power k in the ODEs y' = -z^{k}, z' = y^{k} (creates exponents k+1 and 2k later)",
            "original": 3
          },
          "slot2": {
            "description": "Initial point on the invariant curve",
            "original": "(1, 0)"
          },
          "slot3": {
            "description": "Value C of the conserved quantity y^{k+1}+z^{k+1}=C",
            "original": 1
          },
          "slot4": {
            "description": "Half-constant fraction used in the speed lower-bound argument",
            "original": "1/2"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}