{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "7c3bc808-0b5e-4b31-a7d1-fbcad76c913c",
   "metadata": {},
   "source": [
    "### 超幾何関数と超幾何微分方程式\n",
    "\n",
    "超幾何関数が満たす微分方程式を超幾何微分方程式と呼びます。具体的には\n",
    "$$F(x)=_{2}F_1\\left(a,b;c;x\\right)$$\n",
    "の時、\n",
    "$$x\\,(1-x)\\frac{d^2\\,F(x)}{d\\,x^2}+(c-(a+b+1)\\,x)\\,\\frac{d\\,F(x)}{d\\,x}-a\\,b\\,F(x)=0$$\n",
    "が成り立ちます。これを証明していきます。\n",
    "\n",
    "証明の方針としては$A_n=\\frac{(a)_n\\,(b)_n}{(c)_n}$として、\n",
    "$$F(x)=\\sum_{n=0}^{\\infty}\\frac{A_n\\,x^n}{n!}$$\n",
    "から$\\frac{d}{dx}F(x), \\frac{d^2}{dx^2}F(x)$の級数展開を、係数に$A_{n+1}$が現れる場合と$A_n$が現れる場合と２通り計算します。\n",
    "\n",
    "次に上記の$A_n$の定義から自明に導かれる漸化式$A_{n+1}=\\frac{(a+n)\\,(b+n)}{c+n}\\,A_n$から$(c+n)\\,A_{n+1}=(a+n)\\,(b+n)\\,A_n$の両辺を展開して、さらに両辺の総和を取ります。その各項を先ほど求めた微分に置き換えていくことでこの微分方程式を求めます。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "29f05174-2872-4eb2-a41c-832540426d6d",
   "metadata": {},
   "source": [
    "準備としてout1stTerm()という関数を定義します。総和の式を引数としてとり、総和の最初の項を取り出し、それと第２項以降の総和の和にします。その際に第２項以降の総和を$n=0$から始まるように調整します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "e19e4961-4f1d-41b4-821e-cb58460e4773",
   "metadata": {},
   "outputs": [],
   "source": [
    "out1stTerm(sumexp):=block([ar:args(sumexp),exp,varname,initvalue,finvalue],\n",
    "  [exp,varname,initvalue,finvalue]:ar,\n",
    "  subst(0,varname,exp)+apply(sum,[subst(varname+1,varname,exp),varname,initvalue,finvalue]))$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c90fe632-c6f7-48a0-b1cf-d09f39c2afb0",
   "metadata": {},
   "source": [
    "まず$F(x)$を級数展開の形で定義します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "30db0b79-40c7-4d9b-9608-3c152513b789",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{1}$}F\\left(x\\right)=\\sum_{n=0}^{\\infty }{\\frac{A_{n}\\,x^{n}}{n!}}\\]"
      ],
      "text/plain": [
       "                                     inf       n\n",
       "                                     ====  A  x\n",
       "                                     \\      n\n",
       "(%o1)                         F(x) =  >    -----\n",
       "                                     /      n!\n",
       "                                     ====\n",
       "                                     n = 0"
      ],
      "text/x-maxima": [
       "F(x) = 'sum((A[n]*x^n)/n!,n,0,inf)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F0:F(x)=sum(A[n]/n!*x^n,n,0,inf);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6f21cb6d-708b-4378-b259-1b666199b13d",
   "metadata": {},
   "source": [
    "このまま微分します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "15298aec-7ece-4014-834a-4080a7c29d9d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{2}$}\\frac{d}{d\\,x}\\,F\\left(x\\right)=\\sum_{n=0}^{\\infty }{\\frac{n\\,A_{n}\\,x^{n-1}}{n!}}\\]"
      ],
      "text/plain": [
       "                                     inf         n - 1\n",
       "                                     ====  n A  x\n",
       "                         d           \\        n\n",
       "(%o2)                    -- (F(x)) =  >    -----------\n",
       "                         dx          /         n!\n",
       "                                     ====\n",
       "                                     n = 0"
      ],
      "text/x-maxima": [
       "'diff(F(x),x,1) = 'sum((n*A[n]*x^(n-1))/n!,n,0,inf)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FD1_1:'diff(F(x),x)=diff(rhs(F0),x);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f2dbdc4e-dc1c-498f-ad9e-8dc74bf481de",
   "metadata": {},
   "source": [
    "右辺にout1stTerm()を使って$n=0$の場合を外に出し、第２項以降の総和のインデックスを調整します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "4afd68c3-40de-4342-b7a8-5a5ef7110463",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{3}$}\\frac{d}{d\\,x}\\,F\\left(x\\right)=\\sum_{n=0}^{\\infty }{\\frac{A_{n+1}\\,x^{n}}{n!}}\\]"
      ],
      "text/plain": [
       "                                      inf           n\n",
       "                                      ====  A      x\n",
       "                          d           \\      n + 1\n",
       "(%o3)                     -- (F(x)) =  >    ---------\n",
       "                          dx          /        n!\n",
       "                                      ====\n",
       "                                      n = 0"
      ],
      "text/x-maxima": [
       "'diff(F(x),x,1) = 'sum((A[n+1]*x^n)/n!,n,0,inf)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "SB-KERNEL:REDEFINITION-WITH-DEFUN: redefining MAXIMA::SIMP-UNIT-STEP in DEFUN\n",
      "SB-KERNEL:REDEFINITION-WITH-DEFUN: redefining MAXIMA::SIMP-POCHHAMMER in DEFUN\n"
     ]
    }
   ],
   "source": [
    "FD1_2:lhs(FD1_1)=ev(out1stTerm(rhs(FD1_1)),factorial_expand:true);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "9f276802-360e-4c57-a650-447a46d158c1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{4}$}\\frac{d^2}{d\\,x^2}\\,F\\left(x\\right)=\\sum_{n=0}^{\\infty }{\\frac{\\left(n-1\\right)\\,n\\,A_{n}\\,x^{n-2}}{n!}}\\]"
      ],
      "text/plain": [
       "                                 inf                 n - 2\n",
       "                     2           ====  (n - 1) n A  x\n",
       "                    d            \\                n\n",
       "(%o4)               --- (F(x)) =  >    -------------------\n",
       "                      2          /             n!\n",
       "                    dx           ====\n",
       "                                 n = 0"
      ],
      "text/x-maxima": [
       "'diff(F(x),x,2) = 'sum(((n-1)*n*A[n]*x^(n-2))/n!,n,0,inf)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FD2_1:'diff(F(x),x,2)=diff(rhs(FD1_1),x);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "8b7b1736-d750-43fd-b153-11d073725b2d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{5}$}\\frac{d^2}{d\\,x^2}\\,F\\left(x\\right)=\\sum_{n=0}^{\\infty }{\\frac{n\\,A_{n+1}\\,x^{n-1}}{n!}}\\]"
      ],
      "text/plain": [
       "                                   inf             n - 1\n",
       "                       2           ====  n A      x\n",
       "                      d            \\        n + 1\n",
       "(%o5)                 --- (F(x)) =  >    ---------------\n",
       "                        2          /           n!\n",
       "                      dx           ====\n",
       "                                   n = 0"
      ],
      "text/x-maxima": [
       "'diff(F(x),x,2) = 'sum((n*A[n+1]*x^(n-1))/n!,n,0,inf)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "FD2_2:lhs(FD2_1)=ev(out1stTerm(rhs(FD2_1)),factorial_expand:true);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e019abf4-2153-4963-897b-47e22adb7fcf",
   "metadata": {},
   "source": [
    "$A_n$の定義から次の漸化式が自明に成り立ちます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "407e29be-910f-426c-824f-87da3ad6aa0e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{6}$}A_{n+1}=\\frac{\\left(n+a\\right)\\,\\left(n+b\\right)\\,A_{n}}{n+c}\\]"
      ],
      "text/plain": [
       "                                   (n + a) (n + b) A\n",
       "                                                    n\n",
       "(%o6)                     A      = ------------------\n",
       "                           n + 1         n + c"
      ],
      "text/x-maxima": [
       "A[n+1] = ((n+a)*(n+b)*A[n])/(n+c)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Acond:A[n+1]=(a+n)*(b+n)/(c+n)*A[n];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "e15432bd-71d1-4b12-b67f-f5bdb794f0df",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{7}$}n\\,A_{n+1}+c\\,A_{n+1}=n^2\\,A_{n}+b\\,n\\,A_{n}+a\\,n\\,A_{n}+a\\,b\\,A_{n}\\]"
      ],
      "text/plain": [
       "                                   2\n",
       "(%o7)       n A      + c A      = n  A  + b n A  + a n A  + a b A\n",
       "               n + 1      n + 1       n        n        n        n"
      ],
      "text/x-maxima": [
       "n*A[n+1]+c*A[n+1] = n^2*A[n]+b*n*A[n]+a*n*A[n]+a*b*A[n]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(%*(n+c));"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aee6ebf2-1c74-416d-9d45-6a9d7354ed12",
   "metadata": {},
   "source": [
    "両辺に$\\frac{x^n}{n!}$をかけた式について$n=0$から$\\infty$まで総和を取ります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "a8bbcc0e-2c91-4d6b-bdbc-79633efc31bc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{8}$}\\sum_{n=0}^{\\infty }{\\frac{\\left(n+c\\right)\\,A_{n+1}\\,x^{n}}{n!}}=\\sum_{n=0}^{\\infty }{\\frac{\\left(n^2+\\left(b+a\\right)\\,n+a\\,b\\right)\\,A_{n}\\,x^{n}}{n!}}\\]"
      ],
      "text/plain": [
       "         inf                   n   inf     2                        n\n",
       "         ====  (n + c) A      x    ====  (n  + (b + a) n + a b) A  x\n",
       "         \\              n + 1      \\                             n\n",
       "(%o8)     >    ----------------- =  >    ----------------------------\n",
       "         /            n!           /                  n!\n",
       "         ====                      ====\n",
       "         n = 0                     n = 0"
      ],
      "text/x-maxima": [
       "'sum(((n+c)*A[n+1]*x^n)/n!,n,0,inf) = 'sum(((n^2+(b+a)*n+a*b)*A[n]*x^n)/n!,n,\n",
       "                                           0,inf)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SUM:factor(sum(%*x^n/n!,n,0,inf));"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bde785e6-ce2d-4d79-9d7d-385b48b14a58",
   "metadata": {},
   "source": [
    "上記の式の左辺をじっと睨んでから、FD2_2式の両辺に$x$をかけた式およびFD1_2の両辺に$c$をかけた式の和を計算してみます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "4f35d58c-7498-4172-ba7b-818d33b7f112",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{9}$}x\\,\\left(\\frac{d^2}{d\\,x^2}\\,F\\left(x\\right)\\right)+c\\,\\left(\\frac{d}{d\\,x}\\,F\\left(x\\right)\\right)=\\sum_{n=0}^{\\infty }{\\frac{\\left(n+c\\right)\\,A_{n+1}\\,x^{n}}{n!}}\\]"
      ],
      "text/plain": [
       "                                            inf                   n\n",
       "               2                            ====  (n + c) A      x\n",
       "              d                d            \\              n + 1\n",
       "(%o9)      x (--- (F(x))) + c (-- (F(x))) =  >    -----------------\n",
       "                2              dx           /            n!\n",
       "              dx                            ====\n",
       "                                            n = 0"
      ],
      "text/x-maxima": [
       "x*'diff(F(x),x,2)+c*'diff(F(x),x,1) = 'sum(((n+c)*A[n+1]*x^n)/n!,n,0,inf)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LDE:factor(sumcontract(intosum(FD2_2*x+c*FD1_2)));"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9a282646-d6da-41f5-ba8e-352a8c0d336d",
   "metadata": {},
   "source": [
    "SUMの左辺とLDEの右辺が等しいことが見て取れます。<br>\n",
    "同様にFD2_1に$x^2$, FD1_1に$x\\,(1+a+b)$, F0に$a\\,b$をかけて足した式を計算します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "c50de1e1-0e42-4b0a-88b6-1ed7af8c5248",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{10}$}x^2\\,\\left(\\frac{d^2}{d\\,x^2}\\,F\\left(x\\right)\\right)+b\\,x\\,\\left(\\frac{d}{d\\,x}\\,F\\left(x\\right)\\right)+a\\,x\\,\\left(\\frac{d}{d\\,x}\\,F\\left(x\\right)\\right)+x\\,\\left(\\frac{d}{d\\,x}\\,F\\left(x\\right)\\right)+a\\,b\\,F\\left(x\\right)=\\sum_{n=0}^{\\infty }{\\frac{\\left(n^2+\\left(b+a\\right)\\,n+a\\,b\\right)\\,A_{n}\\,x^{n}}{n!}}\\]"
      ],
      "text/plain": [
       "            2\n",
       "        2  d                  d                 d               d\n",
       "(%o10) x  (--- (F(x))) + b x (-- (F(x))) + a x (-- (F(x))) + x (-- (F(x)))\n",
       "             2                dx                dx              dx\n",
       "           dx\n",
       "                                             inf     2                        n\n",
       "                                             ====  (n  + (b + a) n + a b) A  x\n",
       "                                             \\                             n\n",
       "                                + a b F(x) =  >    ----------------------------\n",
       "                                             /                  n!\n",
       "                                             ====\n",
       "                                             n = 0"
      ],
      "text/x-maxima": [
       "x^2*'diff(F(x),x,2)+b*x*'diff(F(x),x,1)+a*x*'diff(F(x),x,1)+x*'diff(F(x),x,1)\n",
       "                   +a*b*F(x)\n",
       "  = 'sum(((n^2+(b+a)*n+a*b)*A[n]*x^n)/n!,n,0,inf)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RDE:factor(sumcontract(intosum(FD2_1*x^2+FD1_1*x*(1+a+b)+a*b*F0)));"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5072234c-3650-4bc8-8042-3714ea38613f",
   "metadata": {},
   "source": [
    "RDEの右辺とSUMの右辺が等しいことが見て取れます。<br>\n",
    "従ってLDEの左辺とRDEの左辺が等しいことがわかり、次の微分方程式が得られます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "53de7f4b-a2f2-451e-bf63-ec3b3b356285",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{11}$}x\\,\\left(\\frac{d^2}{d\\,x^2}\\,F\\left(x\\right)\\right)+c\\,\\left(\\frac{d}{d\\,x}\\,F\\left(x\\right)\\right)=x^2\\,\\left(\\frac{d^2}{d\\,x^2}\\,F\\left(x\\right)\\right)+b\\,x\\,\\left(\\frac{d}{d\\,x}\\,F\\left(x\\right)\\right)+a\\,x\\,\\left(\\frac{d}{d\\,x}\\,F\\left(x\\right)\\right)+x\\,\\left(\\frac{d}{d\\,x}\\,F\\left(x\\right)\\right)+a\\,b\\,F\\left(x\\right)\\]"
      ],
      "text/plain": [
       "           2\n",
       "          d                d\n",
       "(%o11) x (--- (F(x))) + c (-- (F(x))) = \n",
       "            2              dx\n",
       "          dx\n",
       "      2\n",
       "  2  d                  d                 d               d\n",
       " x  (--- (F(x))) + b x (-- (F(x))) + a x (-- (F(x))) + x (-- (F(x))) + a b F(x)\n",
       "       2                dx                dx              dx\n",
       "     dx"
      ],
      "text/x-maxima": [
       "x*'diff(F(x),x,2)+c*'diff(F(x),x,1) = x^2*'diff(F(x),x,2)\n",
       "                                    +b*x*'diff(F(x),x,1)+a*x*'diff(F(x),x,1)\n",
       "                                    +x*'diff(F(x),x,1)+a*b*F(x)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lhs(LDE)=lhs(RDE);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "5b48d4aa-af41-4add-b709-1627aa0769cf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{12}$}x\\,\\left(\\frac{d^2}{d\\,x^2}\\,F\\left(x\\right)\\right)-x^2\\,\\left(\\frac{d^2}{d\\,x^2}\\,F\\left(x\\right)\\right)=b\\,x\\,\\left(\\frac{d}{d\\,x}\\,F\\left(x\\right)\\right)+a\\,x\\,\\left(\\frac{d}{d\\,x}\\,F\\left(x\\right)\\right)+x\\,\\left(\\frac{d}{d\\,x}\\,F\\left(x\\right)\\right)-c\\,\\left(\\frac{d}{d\\,x}\\,F\\left(x\\right)\\right)+a\\,b\\,F\\left(x\\right)\\]"
      ],
      "text/plain": [
       "           2                 2\n",
       "          d              2  d\n",
       "(%o12) x (--- (F(x))) - x  (--- (F(x))) = \n",
       "            2                 2\n",
       "          dx                dx\n",
       "        d                 d               d               d\n",
       "   b x (-- (F(x))) + a x (-- (F(x))) + x (-- (F(x))) - c (-- (F(x))) + a b F(x)\n",
       "        dx                dx              dx              dx"
      ],
      "text/x-maxima": [
       "x*'diff(F(x),x,2)-x^2*'diff(F(x),x,2) = b*x*'diff(F(x),x,1)\n",
       "                                      +a*x*'diff(F(x),x,1)+x*'diff(F(x),x,1)\n",
       "                                      -c*'diff(F(x),x,1)+a*b*F(x)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%-args(rhs(%))[1]-args(lhs(%))[2];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "16065761-b4f9-4440-9a55-2bdde0a2ce52",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{13}$}\\left(x-x^2\\right)\\,\\left(\\frac{d^2}{d\\,x^2}\\,F\\left(x\\right)\\right)+\\left(\\left(-b-a-1\\right)\\,x+c\\right)\\,\\left(\\frac{d}{d\\,x}\\,F\\left(x\\right)\\right)-a\\,b\\,F\\left(x\\right)=0\\]"
      ],
      "text/plain": [
       "                  2\n",
       "             2   d                                      d\n",
       "(%o13) (x - x ) (--- (F(x))) + (((- b) - a - 1) x + c) (-- (F(x)))\n",
       "                   2                                    dx\n",
       "                 dx\n",
       "                                                                 - a b F(x) = 0"
      ],
      "text/x-maxima": [
       "(x-x^2)*'diff(F(x),x,2)+(((-b)-a-1)*x+c)*'diff(F(x),x,1)-a*b*F(x) = 0"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%-rhs(%),ratsimp;"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4363a68a-7ea2-43b4-858f-c55943a07c0a",
   "metadata": {},
   "source": [
    "少し整理して所望の形の微分方程式を得ることができました。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "2b88aefc-76dd-4876-99ad-16c63714dcee",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{14}$}\\mathbf{true}\\]"
      ],
      "text/plain": [
       "(%o14)                               true"
      ],
      "text/x-maxima": [
       "true"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "display2d;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "2fc31e05-4907-4c46-a63a-b45fc5dc5c55",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{16}$}\\mathbf{false}\\]"
      ],
      "text/plain": [
       "(%o16) false"
      ],
      "text/x-maxima": [
       "false"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "display2d:false;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "0c4a139b-3f2d-41f9-bf3d-57b1e81aeb74",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\(\\left(x-x^2\\right)\\,\\left(\\frac{d^2}{d\\,x^2}\\,F\\left(x\\right)\\right)+\\left(\\left(-b-a-1\\right)\\,x+c\\right)\\,\\left(\\frac{d}{d\\,x}\\,F\\left(x\\right)\\right)-a\\,b\\,F\\left(x\\right)=0\\)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{18}$}\\left(x-x^2\\right)\\,\\left(\\frac{d^2}{d\\,x^2}\\,F\\left(x\\right)\\right)+\\left(\\left(-b-a-1\\right)\\,x+c\\right)\\,\\left(\\frac{d}{d\\,x}\\,F\\left(x\\right)\\right)-a\\,b\\,F\\left(x\\right)=0\\]"
      ],
      "text/plain": [
       "(%o18) (x-x^2)*'diff(F(x),x,2)+(((-b)-a-1)*x+c)*'diff(F(x),x,1)-a*b*F(x) = 0"
      ],
      "text/x-maxima": [
       "(x-x^2)*'diff(F(x),x,2)+(((-b)-a-1)*x+c)*'diff(F(x),x,1)-a*b*F(x) = 0"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(%o13);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "32db64d3-85a4-4afa-a554-590d94ca3f24",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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