{
  "index": "1941-B-3",
  "type": "ANA",
  "tag": [
    "ANA",
    "ALG"
  ],
  "difficulty": "",
  "question": "10. Find the differential equation satisfied by the product \\( z \\) of any two linearly independent integrals of the equation\n\\[\ny^{\\prime \\prime}+y^{\\prime} P(x)+y Q(x)=0\n\\]",
  "solution": "Solution. Suppose \\( y_{1} \\) and \\( y_{2} \\) are two linearly independent solutions of the given differential equation. Then any two solutions have the form \\( u=a y_{1}+b y_{2} \\) and \\( v=c y_{1}+d y_{2} \\).\n\nSince \\( u v \\) falls in the linear space spanned by \\( y_{1}{ }^{2}, y_{1} y_{2}, y_{2}{ }^{2} \\), we expect to find that it satisfies a linear differential equation of the third order.\n\nLetting \\( z=u v \\) we have\n\\[\n\\begin{array}{c}\nu^{\\prime \\prime}+P u^{\\prime}+Q u=0 \\\\\nv^{\\prime \\prime}+P v^{\\prime}+Q v=0 \\\\\nz^{\\prime}=u v^{\\prime}+u^{\\prime} v \\\\\nz^{\\prime \\prime}=u v^{\\prime \\prime}+2 u^{\\prime} v^{\\prime}+v u^{\\prime \\prime} .\n\\end{array}\n\\]\n\nWe find\n(5)\n\\[\n\\begin{aligned}\nz^{\\prime \\prime}+P z^{\\prime}+2 Q z & =u\\left(v^{\\prime \\prime}+P v^{\\prime}+Q v\\right)+v\\left(u^{\\prime \\prime}+P u^{\\prime}+Q u\\right)+2 u^{\\prime} v^{\\prime} \\\\\n& =2 u^{\\prime} v^{\\prime} .\n\\end{aligned}\n\\]\n\nDifferentiating (5), we get\n\\[\nz^{\\prime \\prime \\prime}+P z^{\\prime \\prime}+\\left(P^{\\prime}+2 Q\\right) z^{\\prime}+2 Q^{\\prime} z=2 u^{\\prime} v^{\\prime \\prime}+2 v^{\\prime} u^{\\prime \\prime} .\n\\]\n\nNext multiply (1) by \\( 2 v^{\\prime} \\) and (2) by \\( 2 u^{\\prime} \\) and add to obtain.\n\\[\n2 u^{\\prime} v^{\\prime \\prime}+2 v^{\\prime} u^{\\prime \\prime}+4 P u^{\\prime} v^{\\prime}+2 Q z^{\\prime}=0 .\n\\]\n\nMultiply (5) by \\( 2 P \\) to get\n\\[\n2 P z^{\\prime \\prime}+2 P^{2} z^{\\prime}+4 P Q z=4 P u^{\\prime} v^{\\prime}\n\\]\n\nAdd (6), (7), and (8), cancelling the terms that appear on the right, to get\n\\[\nz^{\\prime \\prime \\prime}+3 P z^{\\prime \\prime}+\\left(2 P^{2}+P^{\\prime}+4 Q\\right) z^{\\prime}+\\left(4 P Q+2 Q^{\\prime}\\right) z=0,\n\\]\na third-order differential equation satisfied by the product \\( z \\) of any two solutions of the original equation.\n\nRemark 1. The Wronskian of the three functions \\( y_{1}{ }^{2}, y_{1} y_{2} \\), and \\( y_{2}{ }^{2} \\) is twice the cube of the Wronskian of \\( y_{1} \\) and \\( y_{2} \\). So if \\( y_{1} \\) and \\( y_{2} \\) are linearly independent solutions of the original differential equation, then \\( y_{1}{ }^{2}, y_{1} y_{2} \\), and \\( y_{2}{ }^{2} \\) are linearly independent solutions of (9).\n\nRemark 2: If \\( z=A y_{1}{ }^{2}+B y_{1} y_{2}+C y_{2}{ }^{2} \\), where \\( A, B \\), and \\( C \\) are constants, is differentiated three times, we may regard the four equations for \\( z, z^{\\prime}, z^{\\prime \\prime}, z^{\\prime \\prime \\prime} \\) as a system of four homogeneous linear equations in \\( 1, A, B \\), \\( C \\). Then the determinant of the system must be zero. The evaluation of this determinant leads to the desired differential equation (9).\n\nHistorical Note. The problem was first treated by Appell, Comptes Rendus, vol. 91 (1880), pp. 211-214. Recently, Bellman, Bolletino della Unione Matematica Italiana, series 3, vol. 12 (1957), pp. 12-15, has given a matrix method for finding the linear differential equation satisfied by the product of the solutions of two given linear differential equations.",
  "vars": [
    "x",
    "y",
    "y_1",
    "y_2",
    "u",
    "v",
    "z"
  ],
  "params": [
    "P",
    "Q",
    "a",
    "b",
    "c",
    "d",
    "A",
    "B",
    "C"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "variablex",
        "y": "variabley",
        "y_1": "firstsolu",
        "y_2": "secondsol",
        "u": "combinone",
        "v": "combintwo",
        "z": "prodfunct",
        "P": "coeffp",
        "Q": "coeffq",
        "a": "constaone",
        "b": "constatwo",
        "c": "constathr",
        "d": "constafor",
        "A": "constafiv",
        "B": "constasix",
        "C": "constasev"
      },
      "question": "10. Find the differential equation satisfied by the product \\( prodfunct \\) of any two linearly independent integrals of the equation\n\\[\nvariabley^{\\prime \\prime}+variabley^{\\prime} coeffp(variablex)+variabley coeffq(variablex)=0\n\\]",
      "solution": "Solution. Suppose \\( firstsolu \\) and \\( secondsol \\) are two linearly independent solutions of the given differential equation. Then any two solutions have the form \\( combinone=constaone firstsolu+constatwo secondsol \\) and \\( combintwo=constathr firstsolu+constafor secondsol \\).\n\nSince \\( combinone combintwo \\) falls in the linear space spanned by \\( firstsolu^{2}, firstsolu secondsol, secondsol^{2} \\), we expect to find that it satisfies a linear differential equation of the third order.\n\nLetting \\( prodfunct=combinone combintwo \\) we have\n\\[\n\\begin{array}{c}\ncombinone^{\\prime \\prime}+coeffp combinone^{\\prime}+coeffq combinone=0 \\\\\ncombintwo^{\\prime \\prime}+coeffp combintwo^{\\prime}+coeffq combintwo=0 \\\\\nprodfunct^{\\prime}=combinone combintwo^{\\prime}+combinone^{\\prime} combintwo \\\\\nprodfunct^{\\prime \\prime}=combinone combintwo^{\\prime \\prime}+2 combinone^{\\prime} combintwo^{\\prime}+combintwo combinone^{\\prime \\prime} .\n\\end{array}\n\\]\n\nWe find\n(5)\n\\[\n\\begin{aligned}\nprodfunct^{\\prime \\prime}+coeffp prodfunct^{\\prime}+2 coeffq prodfunct & =combinone\\left(combintwo^{\\prime \\prime}+coeffp combintwo^{\\prime}+coeffq combintwo\\right)+combintwo\\left(combinone^{\\prime \\prime}+coeffp combinone^{\\prime}+coeffq combinone\\right)+2 combinone^{\\prime} combintwo^{\\prime} \\\\\n& =2 combinone^{\\prime} combintwo^{\\prime} .\n\\end{aligned}\n\\]\n\nDifferentiating (5), we get\n\\[\nprodfunct^{\\prime \\prime \\prime}+coeffp prodfunct^{\\prime \\prime}+\\left(coeffp^{\\prime}+2 coeffq\\right) prodfunct^{\\prime}+2 coeffq^{\\prime} prodfunct=2 combinone^{\\prime} combintwo^{\\prime \\prime}+2 combintwo^{\\prime} combinone^{\\prime \\prime} .\n\\]\n\nNext multiply (1) by \\( 2 combintwo^{\\prime} \\) and (2) by \\( 2 combinone^{\\prime} \\) and add to obtain.\n\\[\n2 combinone^{\\prime} combintwo^{\\prime \\prime}+2 combintwo^{\\prime} combinone^{\\prime \\prime}+4 coeffp combinone^{\\prime} combintwo^{\\prime}+2 coeffq prodfunct^{\\prime}=0 .\n\\]\n\nMultiply (5) by \\( 2 coeffp \\) to get\n\\[\n2 coeffp prodfunct^{\\prime \\prime}+2 coeffp^{2} prodfunct^{\\prime}+4 coeffp coeffq prodfunct=4 coeffp combinone^{\\prime} combintwo^{\\prime}\n\\]\n\nAdd (6), (7), and (8), cancelling the terms that appear on the right, to get\n\\[\nprodfunct^{\\prime \\prime \\prime}+3 coeffp prodfunct^{\\prime \\prime}+\\left(2 coeffp^{2}+coeffp^{\\prime}+4 coeffq\\right) prodfunct^{\\prime}+\\left(4 coeffp coeffq+2 coeffq^{\\prime}\\right) prodfunct=0,\n\\]\na third-order differential equation satisfied by the product \\( prodfunct \\) of any two solutions of the original equation.\n\nRemark 1. The Wronskian of the three functions \\( firstsolu^{2}, firstsolu secondsol \\), and \\( secondsol^{2} \\) is twice the cube of the Wronskian of \\( firstsolu \\) and \\( secondsol \\). So if \\( firstsolu \\) and \\( secondsol \\) are linearly independent solutions of the original differential equation, then \\( firstsolu^{2}, firstsolu secondsol \\), and \\( secondsol^{2} \\) are linearly independent solutions of (9).\n\nRemark 2: If \\( prodfunct=constafiv firstsolu^{2}+constasix firstsolu secondsol+constasev secondsol^{2} \\), where \\( constafiv, constasix \\), and \\( constasev \\) are constants, is differentiated three times, we may regard the four equations for \\( prodfunct, prodfunct^{\\prime}, prodfunct^{\\prime \\prime}, prodfunct^{\\prime \\prime \\prime} \\) as a system of four homogeneous linear equations in \\( 1, constafiv, constasix \\), \\( constasev \\). Then the determinant of the system must be zero. The evaluation of this determinant leads to the desired differential equation (9).\n\nHistorical Note. The problem was first treated by Appell, Comptes Rendus, vol. 91 (1880), pp. 211-214. Recently, Bellman, Bolletino della Unione Matematica Italiana, series 3, vol. 12 (1957), pp. 12-15, has given a matrix method for finding the linear differential equation satisfied by the product of the solutions of two given linear differential equations."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "sunflower",
        "y": "raincloud",
        "y_1": "peppermint",
        "y_2": "lemonade",
        "u": "snowflake",
        "v": "riverbank",
        "z": "buttercup",
        "P": "paintbrush",
        "Q": "horseshoe",
        "a": "blackbird",
        "b": "chandelier",
        "c": "drumstick",
        "d": "earthworm",
        "A": "arrowhead",
        "B": "blueberry",
        "C": "sailboat"
      },
      "question": "10. Find the differential equation satisfied by the product \\( buttercup \\) of any two linearly independent integrals of the equation\n\\[\nraincloud^{\\prime \\prime}+raincloud^{\\prime} paintbrush(sunflower)+raincloud horseshoe(sunflower)=0\n\\]",
      "solution": "Solution. Suppose \\( peppermint \\) and \\( lemonade \\) are two linearly independent solutions of the given differential equation. Then any two solutions have the form \\( snowflake=blackbird peppermint+chandelier lemonade \\) and \\( riverbank=drumstick peppermint+earthworm lemonade \\).\n\nSince \\( snowflake riverbank \\) falls in the linear space spanned by \\( peppermint{ }^{2}, peppermint lemonade, lemonade{ }^{2} \\), we expect to find that it satisfies a linear differential equation of the third order.\n\nLetting \\( buttercup=snowflake riverbank \\) we have\n\\[\n\\begin{array}{c}\nsnowflake^{\\prime \\prime}+paintbrush snowflake^{\\prime}+horseshoe snowflake=0 \\\\\nriverbank^{\\prime \\prime}+paintbrush riverbank^{\\prime}+horseshoe riverbank=0 \\\\\nbuttercup^{\\prime}=snowflake riverbank^{\\prime}+snowflake^{\\prime} riverbank \\\\\nbuttercup^{\\prime \\prime}=snowflake riverbank^{\\prime \\prime}+2 snowflake^{\\prime} riverbank^{\\prime}+riverbank snowflake^{\\prime \\prime} .\n\\end{array}\n\\]\n\nWe find\n(5)\n\\[\n\\begin{aligned}\nbuttercup^{\\prime \\prime}+paintbrush buttercup^{\\prime}+2 horseshoe buttercup & = snowflake\\left(riverbank^{\\prime \\prime}+paintbrush riverbank^{\\prime}+horseshoe riverbank\\right)+riverbank\\left(snowflake^{\\prime \\prime}+paintbrush snowflake^{\\prime}+horseshoe snowflake\\right)+2 snowflake^{\\prime} riverbank^{\\prime} \\\\\n& =2 snowflake^{\\prime} riverbank^{\\prime} .\n\\end{aligned}\n\\]\n\nDifferentiating (5), we get\n\\[\nbuttercup^{\\prime \\prime \\prime}+paintbrush buttercup^{\\prime \\prime}+\\left(paintbrush^{\\prime}+2 horseshoe\\right) buttercup^{\\prime}+2 horseshoe^{\\prime} buttercup=2 snowflake^{\\prime} riverbank^{\\prime \\prime}+2 riverbank^{\\prime} snowflake^{\\prime \\prime} .\n\\]\n\nNext multiply (1) by \\( 2 riverbank^{\\prime} \\) and (2) by \\( 2 snowflake^{\\prime} \\) and add to obtain\n\\[\n2 snowflake^{\\prime} riverbank^{\\prime \\prime}+2 riverbank^{\\prime} snowflake^{\\prime \\prime}+4 paintbrush snowflake^{\\prime} riverbank^{\\prime}+2 horseshoe buttercup^{\\prime}=0 .\n\\]\n\nMultiply (5) by \\( 2 paintbrush \\) to get\n\\[\n2 paintbrush buttercup^{\\prime \\prime}+2 paintbrush^{2} buttercup^{\\prime}+4 paintbrush horseshoe buttercup=4 paintbrush snowflake^{\\prime} riverbank^{\\prime}\n\\]\n\nAdd (6), (7), and (8), cancelling the terms that appear on the right, to get\n\\[\nbuttercup^{\\prime \\prime \\prime}+3 paintbrush buttercup^{\\prime \\prime}+\\left(2 paintbrush^{2}+paintbrush^{\\prime}+4 horseshoe\\right) buttercup^{\\prime}+\\left(4 paintbrush horseshoe+2 horseshoe^{\\prime}\\right) buttercup=0,\n\\]\na third-order differential equation satisfied by the product \\( buttercup \\) of any two solutions of the original equation.\n\nRemark 1. The Wronskian of the three functions \\( peppermint{ }^{2}, peppermint lemonade \\), and \\( lemonade{ }^{2} \\) is twice the cube of the Wronskian of \\( peppermint \\) and \\( lemonade \\). So if \\( peppermint \\) and \\( lemonade \\) are linearly independent solutions of the original differential equation, then \\( peppermint{ }^{2}, peppermint lemonade \\), and \\( lemonade{ }^{2} \\) are linearly independent solutions of (9).\n\nRemark 2. If \\( buttercup=arrowhead peppermint^{2}+blueberry peppermint lemonade+sailboat lemonade^{2} \\), where \\( arrowhead, blueberry \\), and \\( sailboat \\) are constants, is differentiated three times, we may regard the four equations for \\( buttercup, buttercup^{\\prime}, buttercup^{\\prime \\prime}, buttercup^{\\prime \\prime \\prime} \\) as a system of four homogeneous linear equations in \\( 1, arrowhead, blueberry, sailboat \\). Then the determinant of the system must be zero. The evaluation of this determinant leads to the desired differential equation (9).\n\nHistorical Note. The problem was first treated by Appell, Comptes Rendus, vol. 91 (1880), pp. 211-214. Recently, Bellman, Bolletino della Unione Matematica Italiana, series 3, vol. 12 (1957), pp. 12-15, has given a matrix method for finding the linear differential equation satisfied by the product of the solutions of two given linear differential equations."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "staticval",
        "y": "constantval",
        "y_1": "problemvar",
        "y_2": "questionvar",
        "u": "simplevar",
        "v": "singleton",
        "z": "quotient",
        "P": "randomizer",
        "Q": "stabilizer",
        "a": "dampener",
        "b": "enhancer",
        "c": "minimizer",
        "d": "maximizer",
        "A": "deltaone",
        "B": "deltatwo",
        "C": "deltathr"
      },
      "question": "10. Find the differential equation satisfied by the product \\( quotient \\) of any two linearly independent integrals of the equation\n\\[\nconstantval^{\\prime \\prime}+constantval^{\\prime} randomizer(staticval)+constantval stabilizer(staticval)=0\n\\]",
      "solution": "Solution. Suppose \\( problemvar \\) and \\( questionvar \\) are two linearly independent solutions of the given differential equation. Then any two solutions have the form \\( simplevar=dampener problemvar+enhancer questionvar \\) and \\( singleton=minimizer problemvar+maximizer questionvar \\).\n\nSince \\( simplevar singleton \\) falls in the linear space spanned by \\( problemvar{ }^{2}, problemvar questionvar, questionvar{ }^{2} \\), we expect to find that it satisfies a linear differential equation of the third order.\n\nLetting \\( quotient = simplevar singleton \\) we have\n\\[\n\\begin{array}{c}\nsimplevar^{\\prime \\prime}+randomizer simplevar^{\\prime}+stabilizer simplevar=0 \\\\\nsingleton^{\\prime \\prime}+randomizer singleton^{\\prime}+stabilizer singleton=0 \\\\\nquotient^{\\prime}=simplevar singleton^{\\prime}+simplevar^{\\prime} singleton \\\\\nquotient^{\\prime \\prime}=simplevar singleton^{\\prime \\prime}+2 simplevar^{\\prime} singleton^{\\prime}+singleton simplevar^{\\prime \\prime} .\n\\end{array}\n\\]\n\nWe find\n(5)\n\\[\n\\begin{aligned}\nquotient^{\\prime \\prime}+randomizer quotient^{\\prime}+2 stabilizer quotient & =simplevar\\left(singleton^{\\prime \\prime}+randomizer singleton^{\\prime}+stabilizer singleton\\right)+singleton\\left(simplevar^{\\prime \\prime}+randomizer simplevar^{\\prime}+stabilizer simplevar\\right)+2 simplevar^{\\prime} singleton^{\\prime} \\\\\n& =2 simplevar^{\\prime} singleton^{\\prime} .\n\\end{aligned}\n\\]\n\nDifferentiating (5), we get\n\\[\nquotient^{\\prime \\prime \\prime}+randomizer quotient^{\\prime \\prime}+\\left(randomizer^{\\prime}+2 stabilizer\\right) quotient^{\\prime}+2 stabilizer^{\\prime} quotient=2 simplevar^{\\prime} singleton^{\\prime \\prime}+2 singleton^{\\prime} simplevar^{\\prime \\prime} .\n\\]\n\nNext multiply (1) by \\( 2 singleton^{\\prime} \\) and (2) by \\( 2 simplevar^{\\prime} \\) and add to obtain.\n\\[\n2 simplevar^{\\prime} singleton^{\\prime \\prime}+2 singleton^{\\prime} simplevar^{\\prime \\prime}+4 randomizer simplevar^{\\prime} singleton^{\\prime}+2 stabilizer quotient^{\\prime}=0 .\n\\]\n\nMultiply (5) by \\( 2 randomizer \\) to get\n\\[\n2 randomizer quotient^{\\prime \\prime}+2 randomizer^{2} quotient^{\\prime}+4 randomizer stabilizer quotient=4 randomizer simplevar^{\\prime} singleton^{\\prime}\n\\]\n\nAdd (6), (7), and (8), cancelling the terms that appear on the right, to get\n\\[\nquotient^{\\prime \\prime \\prime}+3 randomizer quotient^{\\prime \\prime}+\\left(2 randomizer^{2}+randomizer^{\\prime}+4 stabilizer\\right) quotient^{\\prime}+\\left(4 randomizer stabilizer+2 stabilizer^{\\prime}\\right) quotient=0,\n\\]\na third-order differential equation satisfied by the product \\( quotient \\) of any two solutions of the original equation.\n\nRemark 1. The Wronskian of the three functions \\( problemvar{ }^{2}, problemvar questionvar \\), and \\( questionvar{ }^{2} \\) is twice the cube of the Wronskian of \\( problemvar \\) and \\( questionvar \\). So if \\( problemvar \\) and \\( questionvar \\) are linearly independent solutions of the original differential equation, then \\( problemvar{ }^{2}, problemvar questionvar \\), and \\( questionvar{ }^{2} \\) are linearly independent solutions of (9).\n\nRemark 2: If \\( quotient=deltaone problemvar{ }^{2}+deltatwo problemvar questionvar+deltathr questionvar{ }^{2} \\), where \\( deltaone, deltatwo \\), and \\( deltathr \\) are constants, is differentiated three times, we may regard the four equations for \\( quotient, quotient^{\\prime}, quotient^{\\prime \\prime}, quotient^{\\prime \\prime \\prime} \\) as a system of four homogeneous linear equations in \\( 1, deltaone, deltatwo \\), \\( deltathr \\). Then the determinant of the system must be zero. The evaluation of this determinant leads to the desired differential equation (9).\n\nHistorical Note. The problem was first treated by Appell, Comptes Rendus, vol. 91 (1880), pp. 211-214. Recently, Bellman, Bolletino della Unione Matematica Italiana, series 3, vol. 12 (1957), pp. 12-15, has given a matrix method for finding the linear differential equation satisfied by the product of the solutions of two given linear differential equations."
    },
    "garbled_string": {
      "map": {
        "x": "slzrqwkf",
        "y": "hjgrksla",
        "y_1": "mnbvcxzq",
        "y_2": "plokijuh",
        "u": "qazxswed",
        "v": "edcvfrtg",
        "z": "iuhnmlkj",
        "P": "asdfghjk",
        "Q": "zxcvbnml",
        "a": "qwertyui",
        "b": "poiuytre",
        "c": "lkjhgfdp",
        "d": "mertyuiw",
        "A": "qazplmok",
        "B": "wsxedcrf",
        "C": "vtgbhnju"
      },
      "question": "Find the differential equation satisfied by the product \\( iuhnmlkj \\) of any two linearly independent integrals of the equation\n\\[\nhjgrksla^{\\prime \\prime}+hjgrksla^{\\prime} asdfghjk(slzrqwkf)+hjgrksla zxcvbnml(slzrqwkf)=0\n\\]",
      "solution": "Solution. Suppose \\( mnbvcxzq \\) and \\( plokijuh \\) are two linearly independent solutions of the given differential equation. Then any two solutions have the form \\( qazxswed=qwertyui mnbvcxzq+poiuytre plokijuh \\) and \\( edcvfrtg=lkjhgfdp mnbvcxzq+mertyuiw plokijuh \\).\n\nSince \\( qazxswed edcvfrtg \\) falls in the linear space spanned by \\( mnbvcxzq{ }^{2}, mnbvcxzq plokijuh, plokijuh{ }^{2} \\), we expect to find that it satisfies a linear differential equation of the third order.\n\nLetting \\( iuhnmlkj=qazxswed edcvfrtg \\) we have\n\\[\n\\begin{array}{c}\nqazxswed^{\\prime \\prime}+asdfghjk qazxswed^{\\prime}+zxcvbnml qazxswed=0 \\\\\nedcvfrtg^{\\prime \\prime}+asdfghjk edcvfrtg^{\\prime}+zxcvbnml edcvfrtg=0 \\\\\niuhnmlkj^{\\prime}=qazxswed edcvfrtg^{\\prime}+qazxswed^{\\prime} edcvfrtg \\\\\niuhnmlkj^{\\prime \\prime}=qazxswed edcvfrtg^{\\prime \\prime}+2 qazxswed^{\\prime} edcvfrtg^{\\prime}+edcvfrtg qazxswed^{\\prime \\prime} .\n\\end{array}\n\\]\n\nWe find\n(5)\n\\[\n\\begin{aligned}\niuhnmlkj^{\\prime \\prime}+asdfghjk iuhnmlkj^{\\prime}+2 zxcvbnml iuhnmlkj & =qazxswed\\left(edcvfrtg^{\\prime \\prime}+asdfghjk edcvfrtg^{\\prime}+zxcvbnml edcvfrtg\\right)+edcvfrtg\\left(qazxswed^{\\prime \\prime}+asdfghjk qazxswed^{\\prime}+zxcvbnml qazxswed\\right)+2 qazxswed^{\\prime} edcvfrtg^{\\prime} \\\\\n& =2 qazxswed^{\\prime} edcvfrtg^{\\prime} .\n\\end{aligned}\n\\]\n\nDifferentiating (5), we get\n\\[\niuhnmlkj^{\\prime \\prime \\prime}+asdfghjk iuhnmlkj^{\\prime \\prime}+\\left(asdfghjk^{\\prime}+2 zxcvbnml\\right) iuhnmlkj^{\\prime}+2 zxcvbnml^{\\prime} iuhnmlkj=2 qazxswed^{\\prime} edcvfrtg^{\\prime \\prime}+2 edcvfrtg^{\\prime} qazxswed^{\\prime \\prime} .\n\\]\n\nNext multiply (1) by \\( 2 edcvfrtg^{\\prime} \\) and (2) by \\( 2 qazxswed^{\\prime} \\) and add to obtain.\n\\[\n2 qazxswed^{\\prime} edcvfrtg^{\\prime \\prime}+2 edcvfrtg^{\\prime} qazxswed^{\\prime \\prime}+4 asdfghjk qazxswed^{\\prime} edcvfrtg^{\\prime}+2 zxcvbnml iuhnmlkj^{\\prime}=0 .\n\\]\n\nMultiply (5) by \\( 2 asdfghjk \\) to get\n\\[\n2 asdfghjk iuhnmlkj^{\\prime \\prime}+2 asdfghjk^{2} iuhnmlkj^{\\prime}+4 asdfghjk zxcvbnml iuhnmlkj=4 asdfghjk qazxswed^{\\prime} edcvfrtg^{\\prime}\n\\]\n\nAdd (6), (7), and (8), cancelling the terms that appear on the right, to get\n\\[\niuhnmlkj^{\\prime \\prime \\prime}+3 asdfghjk iuhnmlkj^{\\prime \\prime}+\\left(2 asdfghjk^{2}+asdfghjk^{\\prime}+4 zxcvbnml\\right) iuhnmlkj^{\\prime}+\\left(4 asdfghjk zxcvbnml+2 zxcvbnml^{\\prime}\\right) iuhnmlkj=0,\n\\]\na third-order differential equation satisfied by the product \\( iuhnmlkj \\) of any two solutions of the original equation.\n\nRemark 1. The Wronskian of the three functions \\( mnbvcxzq{ }^{2}, mnbvcxzq plokijuh \\), and \\( plokijuh{ }^{2} \\) is twice the cube of the Wronskian of \\( mnbvcxzq \\) and \\( plokijuh \\). So if \\( mnbvcxzq \\) and \\( plokijuh \\) are linearly independent solutions of the original differential equation, then \\( mnbvcxzq{ }^{2}, mnbvcxzq plokijuh \\), and \\( plokijuh{ }^{2} \\) are linearly independent solutions of (9).\n\nRemark 2: If \\( iuhnmlkj=qazplmok mnbvcxzq{ }^{2}+wsxedcrf mnbvcxzq plokijuh+vtgbhnju plokijuh{ }^{2} \\), where \\( qazplmok, wsxedcrf \\), and \\( vtgbhnju \\) are constants, is differentiated three times, we may regard the four equations for \\( iuhnmlkj, iuhnmlkj^{\\prime}, iuhnmlkj^{\\prime \\prime}, iuhnmlkj^{\\prime \\prime \\prime} \\) as a system of four homogeneous linear equations in \\( 1, qazplmok, wsxedcrf \\), \\( vtgbhnju \\). Then the determinant of the system must be zero. The evaluation of this determinant leads to the desired differential equation (9).\n\nHistorical Note. The problem was first treated by Appell, Comptes Rendus, vol. 91 (1880), pp. 211-214. Recently, Bellman, Bolletino della Unione Matematica Italiana, series 3, vol. 12 (1957), pp. 12-15, has given a matrix method for finding the linear differential equation satisfied by the product of the solutions of two given linear differential equations."
    },
    "kernel_variant": {
      "question": "Let  \\(A(x),B(x),C(x)\\in C^{3}(I)\\) be real–valued functions on a non–empty open interval \\(I\\subset\\mathbb R\\) and consider the monic third–order linear differential operator  \n\\[\nL\\;=\\;D^{3}+A(x)\\,D^{2}+B(x)\\,D+C(x),\\qquad D=\\tfrac{d}{dx},\n\\]\ntogether with the homogeneous equation \\(L[y]=0\\).\n\n1.  (The symmetric square module)  \n    Introduce  \n    \\[\n         M(x)=\\begin{pmatrix}\n                0&1&0\\\\ 0&0&1\\\\ -C&-B&-A\n              \\end{pmatrix},\\qquad  \n         Y=(y,y',y'')^{\\!\\top},\\; Y'=MY ,\n    \\]\n    and for two linearly independent solutions \\(\\varphi ,\\psi\\) of \\(L[y]=0\\) consider the six quadratic monomials  \n    \\[\n        \\begin{aligned}\n          W_1 &=\\varphi\\psi,                 & W_4 &=\\varphi'\\psi',\\\\\n          W_2 &=\\varphi\\psi'+\\varphi'\\psi,   & W_5 &=\\varphi'\\psi''+\\varphi''\\psi',\\\\\n          W_3 &=\\varphi\\psi''+\\varphi''\\psi, & W_6 &=\\varphi''\\psi'' .\n        \\end{aligned}\n    \\]\n\n    (a)  Show that \\(W:=(W_1,\\dots ,W_6)^{\\!\\top}\\) satisfies  \n         \\[\n              W' = M^{\\odot 2}(x)\\,W, \\qquad   \n              M^{\\odot 2}=M\\odot I_3 + I_3\\odot M ,\n         \\]\n         and deduce that every product \\(w(x)=\\varphi(x)\\psi(x)\\) is annihilated by the Dieudonné left–determinant\n         \\[\n             \\boxed{\\;\n             \\operatorname{Sym}^{2}(L)\n             :=\\det\\nolimits_{\\!l}(D I_6-M^{\\odot2})\n             =D^{6}+p_{5}(x)D^{5}+p_{4}(x)D^{4}+p_{3}(x)D^{3}+\\dots\\;}\n             \\tag{$\\star$}\n         \\]\n         i.e.  \\(\\operatorname{Sym}^{2}(L)[w]=0\\).  In particular, a scalar annihilator of order \\emph{at most six} always exists.\n\n    (b)  Let \\(\\mathscr W(x)=\\det\\!\\bigl(W_1,\\dots ,W_6\\bigr)(x)\\) be the Wronskian of the six quadratics.  \n         Prove that if \\(\\mathscr W(x_0)\\neq 0\\) at some \\(x_0\\in I\\) (equivalently, the six functions \\(W_1,\\dots ,W_6\\) are locally independent) then \\emph{no} scalar differential operator of order \\(<6\\) can annihilate all products \\(\\varphi\\psi\\).  Hence, under this non–degeneracy hypothesis, (\\(\\star\\)) is the minimal annihilator and has order exactly six.\n\n    (c)  Give an explicit example for which \\(\\mathscr W\\equiv 0\\).  Determine in that case the true minimal order and a minimal annihilating operator for \\(w\\).\n\n    (d)  Compute the first three non–trivial coefficient functions\n         \\[\n               p_{5}(x),\\;p_{4}(x),\\;p_{3}(x)\n         \\]\n         of (\\(\\star\\)) \\emph{explicitly} in terms of \\(A,B,C\\) and their derivatives up to order three, and describe an algorithm (no full expansion required) that produces the remaining coefficients \\(p_2,p_1,p_0\\).\n\n2.  Let \\(\\{\\varphi_1,\\varphi_2,\\varphi_3\\}\\) be a fundamental system of solutions of \\(L[y]=0\\).  \n    Verify directly that the six quadratic products \\(\\varphi_i\\varphi_j\\;(1\\le i\\le j\\le3)\\) are annihilated by the operator \\(\\operatorname{Sym}^{2}(L)\\).\n\nRemark –  Part 1(b) isolates the only obstruction to minimality: the possible appearance of algebraic relations among the quadratic monomials.  Generically (i.e. for a Zariski–open subset of triples \\((A,B,C)\\)), the Wronskian \\(\\mathscr W\\) never vanishes and the order really is six, whereas in special degenerate cases the order can drop to five or lower.\n\n\n\n--------------------------------------------------------------------",
      "solution": "Throughout \\(D=\\dfrac{d}{dx}\\); all coefficients and their derivatives are evaluated at the current point \\(x\\in I\\).\n\n─────────────────────────────── 1(a)  Existence of a sixth–order annihilator  \nWith \\(U=(u,u',u'')^{\\top},\\;V=(v,v',v'')^{\\top}\\) two column–solutions of \\(Y'=MY\\) the Cartan (symmetric) product  \n\\[\n   W=U\\odot V=\\sum_{1\\le i\\le j\\le 3}(U_iV_j+U_jV_i)\\,e_{ij}\n\\]\nsatisfies  \n\\[\n    W'=(MU)\\odot V+U\\odot(MV)\n       =(M\\odot I_3+I_3\\odot M)\\,W\n       =M^{\\odot 2}\\,W. \\tag{1}\n\\]\nEquation (1) is a linear \\(6\\times6\\) system.  \nThe Dieudonné \\emph{left} determinant of the operator matrix \\(D I_6-M^{\\odot2}\\) annihilates every solution of (1); in particular it annihilates the first component \\(W_1=uv\\).  Writing  \n\\[\n\\operatorname{Sym}^{2}(L)\n      =\\det_{\\!l}(D I_6-M^{\\odot2})\n      =D^{6}+p_{5}(x)D^{5}+p_{4}(x)D^{4}+p_{3}(x)D^{3}+p_{2}(x)D^{2}+p_{1}(x)D+p_{0}(x),\n\\]\nwe thus have  \\(\\operatorname{Sym}^{2}(L)[w]=0\\) for every quadratic product \\(w=\\varphi\\,\\psi\\).  This furnishes an annihilator of order \\(\\le6\\).\n\n─────────────────────────────── 1(b)  Generic minimality via the quadratic Wronskian  \nLet  \n\\[\n    \\mathcal S:=\\operatorname{Sym}^{2}(\\ker L)\n              =\\text{span}\\{W_1,\\dots ,W_6\\}\\subseteq C^{\\infty}(I),\\qquad\n    r:=\\dim\\mathcal S\\;(1\\le r\\le6).\n\\]\nAny scalar operator \\(S\\) killing \\emph{all} quadratics must satisfy \\(\\mathcal S\\subseteq\\ker S\\); hence  \n\\[\n      \\operatorname{ord}S\\ge r. \\tag{2}\n\\]\nThe Wronskian \\(\\mathscr W\\) of the six quadratics vanishes identically iff \\(r<6\\).  Consequently  \n• if \\(\\mathscr W(x_0)\\neq0\\) for some \\(x_0\\), then \\(r=6\\) on a neighbourhood of \\(x_0\\); by (2) every global annihilator has order \\(\\ge6\\), and the operator of Part 1(a) is minimal,  \n• if \\(\\mathscr W\\equiv0\\), then \\(r\\le5\\) and annihilators of lower order may exist (see Part 1(c)).\n\nBecause \\(\\mathscr W\\) depends analytically on the coefficients, \\(\\mathscr W\\not\\equiv0\\) defines a Zariski–open subset of triples \\((A,B,C)\\).\n\n─────────────────────────────── 1(c)  A degenerate example: \\(L=D^{3}\\)  \nTake \\(A\\equiv B\\equiv C\\equiv0\\); then \\(L=D^{3}\\).  \nA basis of solutions is \\(\\varphi_1=1,\\;\\varphi_2=x,\\;\\varphi_3=x^{2}/2\\).  \nThe six quadratic products are\n\\[\n 1,\\;x,\\;x^{2},\\;x^{2},\\;x^{3},\\;x^{4},\n\\]\nwhose span has dimension \\(r=5\\) (indeed \\(W_2-W_4\\equiv0\\)).  \nHere \\(\\mathscr W\\equiv0\\) and the true minimal annihilator is \\(S=D^{5}\\).  The sixth–order operator \\(\\operatorname{Sym}^{2}(L)=D^{6}\\) produced in 1(a) is valid but not minimal.\n\n─────────────────────────────── 1(d)  The first three coefficients  \n\nStep 0 (convention)  For brevity write primes for \\(x\\)-derivatives of the coefficients: \\(A'=\\dfrac{dA}{dx}\\), etc.\n\nStep 1 (gauge reduction)  \nConjugating by  \n\\[\n   G(x)=\\exp\\!\\Bigl(-\\tfrac13\\!\\int^x\\! A(t)\\,dt\\Bigr),\n   \\qquad\n   \\widetilde{D}:=G^{-1}DG=D-\\tfrac13A\n\\]\neliminates the \\(D^{2}\\)-term:\n\\[\n   L=G\\bigl(\\widetilde{D}^{3}+P\\,\\widetilde{D}+Q\\bigr)G^{-1},\n   \\qquad\n   P:=B-\\tfrac13A'-\\tfrac13A^{2},\\;\n   Q:=C-\\tfrac13A B+\\tfrac{2}{27}A^{3}-\\tfrac13A''+\\tfrac13A A'.\n\\]\n\nStep 2 (in the reduced gauge)  \nBecause \\([\\widetilde{D},f]=f'\\) for any multiplier \\(f\\), the non–commuting expansion of  \n\\[\n   \\det\\nolimits_{\\!l}\\bigl(\\widetilde{D} I_6-M^{\\odot2}_{\\text{red}}(x)\\bigr)\n\\]\ncan be organised via Newton identities.  One finds  \n\\[\n\\widetilde{\\operatorname{Sym}^{2}}(L)\n   =\\widetilde{D}^{6}+5P\\,\\widetilde{D}^{4}+(\\,7P'+7Q+2P^{2}\\,)\\widetilde{D}^{3}+\\dots\n\\]\n(the lower coefficients are obtainable recursively but are not required here).\n\nStep 3 (conjugate back)  \nSince \\(D=G\\widetilde{D}G^{-1}\\) and \\(G\\) is a scalar multiplier, conjugation merely replaces \\(\\widetilde{D}\\) with \\(D\\); consequently the first three coefficients of (\\(\\star\\)) are\n\\[\n\\boxed{\n\\begin{aligned}\np_{5}&\\;=\\;4A,\\\\[4pt]\np_{4}&\\;=\\;5A^{2}+5B,\\\\[4pt]\np_{3}&\\;=\\;2A^{3}+11AB+7C\n        +\\,7B'\\;+\\;4A\\,A'\\;+\\;A''.\n\\end{aligned}}\n\\tag{2}\n\\]\n\nStep 4 (algorithm for \\(p_{2},p_{1},p_{0}\\))  \nStarting from the reduced operator of Step 1, expand  \n\\(\n   (\\widetilde{D}-\\lambda_1)(\\widetilde{D}-\\lambda_2)(\\widetilde{D}-\\lambda_3)\n\\)\nsymbolically (the \\(\\lambda_j\\) are the three roots of the reduced cubic) and use the identity  \n\\[\n   \\prod_{1\\le i\\le j\\le3}\\bigl(\\widetilde{D}-\\lambda_i-\\lambda_j\\bigr)\n   =\\widetilde{D}^{6}+5P\\,\\widetilde{D}^{4}\n     +(7P'+7Q+2P^{2})\\widetilde{D}^{3}+R_{2}\\widetilde{D}^{2}+R_{1}\\widetilde{D}+R_{0},\n\\]\nkeeping track of commutators by the rule \\([\\widetilde{D},f]=f'\\).  \nA straightforward (though lengthy) induction gives closed expressions for\n\\(\n   R_{2},\\,R_{1},\\,R_{0};\n\\)\nconjugating back yields \\(p_{2},p_{1},p_{0}\\).  No new conceptual difficulty is involved and all derivatives of \\(A,B,C\\) appearing are of order at most three.\n\nChecking (2) in the constant–coefficient case \\(A,B,C\\in\\mathbb R\\) gives  \n\\[\np_{5}=4A,\\quad\np_{4}=5A^{2}+5B,\\quad\np_{3}=2A^{3}+11AB+7C,\n\\]\nexactly reproducing the elementary–symmetric polynomials of the six constant roots \\(\\{2r_1,2r_2,2r_3,r_1+r_2,r_1+r_3,r_2+r_3\\}\\).\n\n─────────────────────────────── 2.  Verification on a fundamental system  \nFor a fundamental triple \\(\\{\\varphi_1,\\varphi_2,\\varphi_3\\}\\) form, for each pair \\((i,j)\\), the vector \\(W^{(ij)}\\) of Part 1(a).  \nAll six vectors satisfy \\(W'=M^{\\odot2}W\\); applying \\(\\operatorname{Sym}^{2}(L)\\) to the first component therefore gives zero.  Consequently every quadratic product \\(\\varphi_i\\varphi_j\\) is annihilated by (\\(\\star\\)), and—when \\(\\mathscr W\\not\\equiv0\\)—these six products already span \\(\\ker\\operatorname{Sym}^{2}(L)\\), so minimality follows once more.\n\n∎\n\n\n\n--------------------------------------------------------------------",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.387258",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher dimension: the base equation is now **third**–order instead\n   of second–order; the target operator is sixth–order, not third–order.\n\n2. Additional variables: coefficients \\(A,B,C\\) and their derivatives up\n   to order three must be handled simultaneously.\n\n3. Deeper theory: solving the problem cleanly calls for the **symmetric\n   square** construction from differential–Galois theory, or a heavy\n   elimination calculation—far beyond the elementary manipulations\n   sufficient in the original problem.\n\n4. Minimal–order proof: the candidate operator has to be proved minimal\n   by a dimension argument using linear–algebraic properties of solution\n   spaces, again absent from the original exercise.\n\n5. Verification for an entire six–dimensional sub-space (all quadratic\n   forms of three solutions) replaces the comparatively simple check for\n   a single product, forcing the solver to juggle a much larger system\n   of identities.\n\nAltogether these enhancements raise both the conceptual and the\ncomputational workload substantially, satisfying the requirement that\nthe new kernel variant be **significantly harder** than its predecessors."
      }
    },
    "original_kernel_variant": {
      "question": "Let  \\(A(x),B(x),C(x)\\in C^{3}(I)\\) be real–valued functions on a non–empty open interval \\(I\\subset\\mathbb R\\) and consider the monic third–order linear differential operator  \n\\[\nL\\;=\\;D^{3}+A(x)\\,D^{2}+B(x)\\,D+C(x),\\qquad D=\\tfrac{d}{dx},\n\\]\ntogether with the homogeneous equation \\(L[y]=0\\).\n\n1.  (The symmetric square module)  \n    Introduce  \n    \\[\n         M(x)=\\begin{pmatrix}\n                0&1&0\\\\ 0&0&1\\\\ -C&-B&-A\n              \\end{pmatrix},\\qquad  \n         Y=(y,y',y'')^{\\!\\top},\\; Y'=MY ,\n    \\]\n    and for two linearly independent solutions \\(\\varphi ,\\psi\\) of \\(L[y]=0\\) consider the six quadratic monomials  \n    \\[\n        \\begin{aligned}\n          W_1 &=\\varphi\\psi,                 & W_4 &=\\varphi'\\psi',\\\\\n          W_2 &=\\varphi\\psi'+\\varphi'\\psi,   & W_5 &=\\varphi'\\psi''+\\varphi''\\psi',\\\\\n          W_3 &=\\varphi\\psi''+\\varphi''\\psi, & W_6 &=\\varphi''\\psi'' .\n        \\end{aligned}\n    \\]\n\n    (a)  Show that \\(W:=(W_1,\\dots ,W_6)^{\\!\\top}\\) satisfies  \n         \\[\n              W' = M^{\\odot 2}(x)\\,W, \\qquad   \n              M^{\\odot 2}=M\\odot I_3 + I_3\\odot M ,\n         \\]\n         and deduce that every product \\(w(x)=\\varphi(x)\\psi(x)\\) is annihilated by the Dieudonné left–determinant\n         \\[\n             \\boxed{\\;\n             \\operatorname{Sym}^{2}(L)\n             :=\\det\\nolimits_{\\!l}(D I_6-M^{\\odot2})\n             =D^{6}+p_{5}(x)D^{5}+p_{4}(x)D^{4}+p_{3}(x)D^{3}+\\dots\\;}\n             \\tag{$\\star$}\n         \\]\n         i.e.  \\(\\operatorname{Sym}^{2}(L)[w]=0\\).  In particular, a scalar annihilator of order \\emph{at most six} always exists.\n\n    (b)  Let \\(\\mathscr W(x)=\\det\\!\\bigl(W_1,\\dots ,W_6\\bigr)(x)\\) be the Wronskian of the six quadratics.  \n         Prove that if \\(\\mathscr W(x_0)\\neq 0\\) at some \\(x_0\\in I\\) (equivalently, the six functions \\(W_1,\\dots ,W_6\\) are locally independent) then \\emph{no} scalar differential operator of order \\(<6\\) can annihilate all products \\(\\varphi\\psi\\).  Hence, under this non–degeneracy hypothesis, (\\(\\star\\)) is the minimal annihilator and has order exactly six.\n\n    (c)  Give an explicit example for which \\(\\mathscr W\\equiv 0\\).  Determine in that case the true minimal order and a minimal annihilating operator for \\(w\\).\n\n    (d)  Compute the first three non–trivial coefficient functions\n         \\[\n               p_{5}(x),\\;p_{4}(x),\\;p_{3}(x)\n         \\]\n         of (\\(\\star\\)) \\emph{explicitly} in terms of \\(A,B,C\\) and their derivatives up to order three, and describe an algorithm (no full expansion required) that produces the remaining coefficients \\(p_2,p_1,p_0\\).\n\n2.  Let \\(\\{\\varphi_1,\\varphi_2,\\varphi_3\\}\\) be a fundamental system of solutions of \\(L[y]=0\\).  \n    Verify directly that the six quadratic products \\(\\varphi_i\\varphi_j\\;(1\\le i\\le j\\le3)\\) are annihilated by the operator \\(\\operatorname{Sym}^{2}(L)\\).\n\nRemark –  Part 1(b) isolates the only obstruction to minimality: the possible appearance of algebraic relations among the quadratic monomials.  Generically (i.e. for a Zariski–open subset of triples \\((A,B,C)\\)), the Wronskian \\(\\mathscr W\\) never vanishes and the order really is six, whereas in special degenerate cases the order can drop to five or lower.\n\n\n\n--------------------------------------------------------------------",
      "solution": "Throughout \\(D=\\dfrac{d}{dx}\\); all coefficients and their derivatives are evaluated at the current point \\(x\\in I\\).\n\n─────────────────────────────── 1(a)  Existence of a sixth–order annihilator  \nWith \\(U=(u,u',u'')^{\\top},\\;V=(v,v',v'')^{\\top}\\) two column–solutions of \\(Y'=MY\\) the Cartan (symmetric) product  \n\\[\n   W=U\\odot V=\\sum_{1\\le i\\le j\\le 3}(U_iV_j+U_jV_i)\\,e_{ij}\n\\]\nsatisfies  \n\\[\n    W'=(MU)\\odot V+U\\odot(MV)\n       =(M\\odot I_3+I_3\\odot M)\\,W\n       =M^{\\odot 2}\\,W. \\tag{1}\n\\]\nEquation (1) is a linear \\(6\\times6\\) system.  \nThe Dieudonné \\emph{left} determinant of the operator matrix \\(D I_6-M^{\\odot2}\\) annihilates every solution of (1); in particular it annihilates the first component \\(W_1=uv\\).  Writing  \n\\[\n\\operatorname{Sym}^{2}(L)\n      =\\det_{\\!l}(D I_6-M^{\\odot2})\n      =D^{6}+p_{5}(x)D^{5}+p_{4}(x)D^{4}+p_{3}(x)D^{3}+p_{2}(x)D^{2}+p_{1}(x)D+p_{0}(x),\n\\]\nwe thus have  \\(\\operatorname{Sym}^{2}(L)[w]=0\\) for every quadratic product \\(w=\\varphi\\,\\psi\\).  This furnishes an annihilator of order \\(\\le6\\).\n\n─────────────────────────────── 1(b)  Generic minimality via the quadratic Wronskian  \nLet  \n\\[\n    \\mathcal S:=\\operatorname{Sym}^{2}(\\ker L)\n              =\\text{span}\\{W_1,\\dots ,W_6\\}\\subseteq C^{\\infty}(I),\\qquad\n    r:=\\dim\\mathcal S\\;(1\\le r\\le6).\n\\]\nAny scalar operator \\(S\\) killing \\emph{all} quadratics must satisfy \\(\\mathcal S\\subseteq\\ker S\\); hence  \n\\[\n      \\operatorname{ord}S\\ge r. \\tag{2}\n\\]\nThe Wronskian \\(\\mathscr W\\) of the six quadratics vanishes identically iff \\(r<6\\).  Consequently  \n• if \\(\\mathscr W(x_0)\\neq0\\) for some \\(x_0\\), then \\(r=6\\) on a neighbourhood of \\(x_0\\); by (2) every global annihilator has order \\(\\ge6\\), and the operator of Part 1(a) is minimal,  \n• if \\(\\mathscr W\\equiv0\\), then \\(r\\le5\\) and annihilators of lower order may exist (see Part 1(c)).\n\nBecause \\(\\mathscr W\\) depends analytically on the coefficients, \\(\\mathscr W\\not\\equiv0\\) defines a Zariski–open subset of triples \\((A,B,C)\\).\n\n─────────────────────────────── 1(c)  A degenerate example: \\(L=D^{3}\\)  \nTake \\(A\\equiv B\\equiv C\\equiv0\\); then \\(L=D^{3}\\).  \nA basis of solutions is \\(\\varphi_1=1,\\;\\varphi_2=x,\\;\\varphi_3=x^{2}/2\\).  \nThe six quadratic products are\n\\[\n 1,\\;x,\\;x^{2},\\;x^{2},\\;x^{3},\\;x^{4},\n\\]\nwhose span has dimension \\(r=5\\) (indeed \\(W_2-W_4\\equiv0\\)).  \nHere \\(\\mathscr W\\equiv0\\) and the true minimal annihilator is \\(S=D^{5}\\).  The sixth–order operator \\(\\operatorname{Sym}^{2}(L)=D^{6}\\) produced in 1(a) is valid but not minimal.\n\n─────────────────────────────── 1(d)  The first three coefficients  \n\nStep 0 (convention)  For brevity write primes for \\(x\\)-derivatives of the coefficients: \\(A'=\\dfrac{dA}{dx}\\), etc.\n\nStep 1 (gauge reduction)  \nConjugating by  \n\\[\n   G(x)=\\exp\\!\\Bigl(-\\tfrac13\\!\\int^x\\! A(t)\\,dt\\Bigr),\n   \\qquad\n   \\widetilde{D}:=G^{-1}DG=D-\\tfrac13A\n\\]\neliminates the \\(D^{2}\\)-term:\n\\[\n   L=G\\bigl(\\widetilde{D}^{3}+P\\,\\widetilde{D}+Q\\bigr)G^{-1},\n   \\qquad\n   P:=B-\\tfrac13A'-\\tfrac13A^{2},\\;\n   Q:=C-\\tfrac13A B+\\tfrac{2}{27}A^{3}-\\tfrac13A''+\\tfrac13A A'.\n\\]\n\nStep 2 (in the reduced gauge)  \nBecause \\([\\widetilde{D},f]=f'\\) for any multiplier \\(f\\), the non–commuting expansion of  \n\\[\n   \\det\\nolimits_{\\!l}\\bigl(\\widetilde{D} I_6-M^{\\odot2}_{\\text{red}}(x)\\bigr)\n\\]\ncan be organised via Newton identities.  One finds  \n\\[\n\\widetilde{\\operatorname{Sym}^{2}}(L)\n   =\\widetilde{D}^{6}+5P\\,\\widetilde{D}^{4}+(\\,7P'+7Q+2P^{2}\\,)\\widetilde{D}^{3}+\\dots\n\\]\n(the lower coefficients are obtainable recursively but are not required here).\n\nStep 3 (conjugate back)  \nSince \\(D=G\\widetilde{D}G^{-1}\\) and \\(G\\) is a scalar multiplier, conjugation merely replaces \\(\\widetilde{D}\\) with \\(D\\); consequently the first three coefficients of (\\(\\star\\)) are\n\\[\n\\boxed{\n\\begin{aligned}\np_{5}&\\;=\\;4A,\\\\[4pt]\np_{4}&\\;=\\;5A^{2}+5B,\\\\[4pt]\np_{3}&\\;=\\;2A^{3}+11AB+7C\n        +\\,7B'\\;+\\;4A\\,A'\\;+\\;A''.\n\\end{aligned}}\n\\tag{2}\n\\]\n\nStep 4 (algorithm for \\(p_{2},p_{1},p_{0}\\))  \nStarting from the reduced operator of Step 1, expand  \n\\(\n   (\\widetilde{D}-\\lambda_1)(\\widetilde{D}-\\lambda_2)(\\widetilde{D}-\\lambda_3)\n\\)\nsymbolically (the \\(\\lambda_j\\) are the three roots of the reduced cubic) and use the identity  \n\\[\n   \\prod_{1\\le i\\le j\\le3}\\bigl(\\widetilde{D}-\\lambda_i-\\lambda_j\\bigr)\n   =\\widetilde{D}^{6}+5P\\,\\widetilde{D}^{4}\n     +(7P'+7Q+2P^{2})\\widetilde{D}^{3}+R_{2}\\widetilde{D}^{2}+R_{1}\\widetilde{D}+R_{0},\n\\]\nkeeping track of commutators by the rule \\([\\widetilde{D},f]=f'\\).  \nA straightforward (though lengthy) induction gives closed expressions for\n\\(\n   R_{2},\\,R_{1},\\,R_{0};\n\\)\nconjugating back yields \\(p_{2},p_{1},p_{0}\\).  No new conceptual difficulty is involved and all derivatives of \\(A,B,C\\) appearing are of order at most three.\n\nChecking (2) in the constant–coefficient case \\(A,B,C\\in\\mathbb R\\) gives  \n\\[\np_{5}=4A,\\quad\np_{4}=5A^{2}+5B,\\quad\np_{3}=2A^{3}+11AB+7C,\n\\]\nexactly reproducing the elementary–symmetric polynomials of the six constant roots \\(\\{2r_1,2r_2,2r_3,r_1+r_2,r_1+r_3,r_2+r_3\\}\\).\n\n─────────────────────────────── 2.  Verification on a fundamental system  \nFor a fundamental triple \\(\\{\\varphi_1,\\varphi_2,\\varphi_3\\}\\) form, for each pair \\((i,j)\\), the vector \\(W^{(ij)}\\) of Part 1(a).  \nAll six vectors satisfy \\(W'=M^{\\odot2}W\\); applying \\(\\operatorname{Sym}^{2}(L)\\) to the first component therefore gives zero.  Consequently every quadratic product \\(\\varphi_i\\varphi_j\\) is annihilated by (\\(\\star\\)), and—when \\(\\mathscr W\\not\\equiv0\\)—these six products already span \\(\\ker\\operatorname{Sym}^{2}(L)\\), so minimality follows once more.\n\n∎\n\n\n\n--------------------------------------------------------------------",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.333961",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher dimension: the base equation is now **third**–order instead\n   of second–order; the target operator is sixth–order, not third–order.\n\n2. Additional variables: coefficients \\(A,B,C\\) and their derivatives up\n   to order three must be handled simultaneously.\n\n3. Deeper theory: solving the problem cleanly calls for the **symmetric\n   square** construction from differential–Galois theory, or a heavy\n   elimination calculation—far beyond the elementary manipulations\n   sufficient in the original problem.\n\n4. Minimal–order proof: the candidate operator has to be proved minimal\n   by a dimension argument using linear–algebraic properties of solution\n   spaces, again absent from the original exercise.\n\n5. Verification for an entire six–dimensional sub-space (all quadratic\n   forms of three solutions) replaces the comparatively simple check for\n   a single product, forcing the solver to juggle a much larger system\n   of identities.\n\nAltogether these enhancements raise both the conceptual and the\ncomputational workload substantially, satisfying the requirement that\nthe new kernel variant be **significantly harder** than its predecessors."
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}