SymPy_Example.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "from sympy.matrices import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sympy.abc import x, F, f, u, l, a, k, y, delta\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x, F, f, u, l, a, k, y, delta_x= symbols(\"x F f u l a k y delta_x\", real = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "F =u**2/2+l*ln((a-u))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGoAAAArBAMAAABiCwBtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMnZmme8iq7uJ\nRFQp1WXSAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABTUlEQVRIDWNgwAJY7R2wiBIUyieoApuCImyC\nBMVaCarAooDzCxZBgkJsHwgqwaKAawMWQYJCHAEElWBR4O+ARZCgUARBFdgU7GLEJkpI7BsPIRXY\n5P+KYhMlJJZ+gJCKwSj/lCxHDYAuxvAAhhkCRLsW6kJ2vgaGeKI1MUB1TWNXYNjNwCBkDAIq+LUf\nUlKyUFLSAioSAOaiViTF/3EAaL6GheF9ARYSygeYrmIGVqg5SDbiZMJ07WLg3pCIUxW6BExXNcON\nggPokjj5MF1SR58sdcCpCl0CpgtdHD9fEr/0sJPlrCltRPEUUbnixgQm1LRCTK5gbGNg34BiFzRX\noIihc9g/MjAHAAUZlUFZx8gBM1eg6wDxOTYwyAOVIgMicoX8AYb3yFqAbCJyhfwDhk18E1D0EZEr\nmALYrNDqEiJyBePSI7NiUaxiwJkrAKlwXkCyHtZxAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\frac{l}{a - u} + u$$"
      ],
      "text/plain": [
       "    l      \n",
       "- ───── + u\n",
       "  a - u    "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = diff(F,u)\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKoAAAAvBAMAAACBJUzaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/QcmS7SbKwy9LwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$l \\log{\\left (- a + u \\right )} + \\frac{u^{2}}{2}$$"
      ],
      "text/plain": [
       "                 2\n",
       "                u \n",
       "l⋅log(-a + u) + ──\n",
       "                2 "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PE = integrate(f, u)\n",
    "PE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUIAAAAyBAMAAADb1s/aAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n3SnaRX908r/h+Lf/b9jD5v/rbdP/Bfwn7XOsUUnkPMEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ \\frac{a}{2} - \\frac{1}{2} \\sqrt{a^{2} - 4 l}, \\quad \\frac{a}{2} + \\frac{1}{2} \\sqrt{a^{2} - 4 l}\\right ]$$"
      ],
      "text/plain": [
       "⎡       __________         __________⎤\n",
       "⎢      ╱  2               ╱  2       ⎥\n",
       "⎢a   ╲╱  a  - 4⋅l   a   ╲╱  a  - 4⋅l ⎥\n",
       "⎢─ - ─────────────, ─ + ─────────────⎥\n",
       "⎣2         2        2         2      ⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(f,u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEUAAAAyBAMAAAD4hAtjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTT3fpjq+cT88FXg3axF4cQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}k x y\\\\x^{2} - y\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡k⋅x⋅y ⎤\n",
       "⎢      ⎥\n",
       "⎢ 2    ⎥\n",
       "⎣x  - y⎦"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F = Matrix([[k*x*y],[-y+x**2]])\n",
    "F"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAyBAMAAAC0bgabAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhB2qyJEZt3N\nu5lcG7FTAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyElEQVQoFWOQ//+JAQz0/39kEHZxhXBCXAwZ\nRCBMEOlIHU5YemdSAtQ0VgWubdwHoBymAKavzA0wGQaeC0A7YZbyJyBx/B3gHOaAfAa2AKiy+Qb3\nGbqAUmCHhqeFvVoA44DUgwCVvAAxbABNY81LYGiEeZuJWYAhH+a5Zr4DDHdhnADGBIZvMA7D/ACW\nL3DOGwY2ATjnOgNvAZzzlmH+AjgnXO19A5zDwPAGyIZ4m/0A6084h3sBbwKcw5n2BMTGGzooSQw5\n8QEAD987LRSrD1UAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}x\\\\y\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡x⎤\n",
       "⎢ ⎥\n",
       "⎣y⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = Matrix([[x],[y]])\n",
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF4AAAAyBAMAAAAnynF8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhCZZs3dIrur\nRHbLQ9+lAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACZ0lEQVRIDe2Wz2sTQRTHv9nsZtMk2xSsiCeD\nKMVLqn+AOEIVTyYV8QcWswdvLRjQS08G/IW3RU9ChZw8eLA5CB4bQdCTFK8FU4RGqDVGm4AINc7O\nbjpvdpOwPXhzDjvf997nPWbesMPgUO8noo7p3g/smzkTFcfFmROYHE1/uq3ET0v+0tQRJSQM8+QK\nE+LaQTER/i6uVoWPfjIwfPOBmCVvTkCrUFRordx3bQsh+XQd8W4/uDtbv31peTHJZ+uwvBq7sCse\n+ZbmCCH5kg3rjx+U08vvOc9I2vEtBkh+OQc9dHL6/XTb40vHNxt8L4S3B/CzzOx4/KnrxctcSX7g\neuah+/xNW+RJnu83HtrvAeh+z5a+rrsJkk/XYAb7aXRgOsjf2GLGTmpH5fl5JSqui44O0uXx4vux\nVbNtbGs8IuvjCWYZZV2dxxp0PAQyNXQC/IXW2yAOrWVzHz/jdBnNDS5J/RDcd+htvS+j8Mua467E\nGxHqz33LH+3jUervsq6IUP8/73fA+sKUXgwyvH4+n3/Fg7fW/H9jEOj7BG/UMb0K1FEcTsb3i5jg\nM0XEJoCp4TTmFsj9E3OQ4UtpVEckjBM+2RV80onIc2ysi1QrdP2QfFqfuxs5nCuukHhQBvglGJO4\nw4KUtFU+YSNRQyHMX3nqjnuAyjeBWBmFEf1X+LiNFyWGRWjN8+FLQixK4c8Cm1mG19jIVJ7JNVNF\n+dTjwwuOtm7VwJJlChFN+WSv1+PXJN8EslXCEJl48+udawb/9wIYocJS5VPtY8iFIeJReePDx88j\nesrzVJ4UGiL/Pb/X98we30t/ASoRn12Eb7DlAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & \\delta_{x} k\\\\2 \\delta_{x} & -1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ 0    δₓ⋅k⎤\n",
       "⎢          ⎥\n",
       "⎣2⋅δₓ   -1 ⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fm = F.jacobian(X).subs({x:delta_x,y:0})\n",
    "Fm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAbQAAAAyBAMAAADGhDMEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left \\{ - \\frac{1}{2} \\sqrt{8 \\delta_{x}^{2} k + 1} - \\frac{1}{2} : 1, \\quad \\frac{1}{2} \\sqrt{8 \\delta_{x}^{2} k + 1} - \\frac{1}{2} : 1\\right \\}$$"
      ],
      "text/plain": [
       "⎧     _____________            _____________       ⎫\n",
       "⎪    ╱     2                  ╱     2              ⎪\n",
       "⎨  ╲╱  8⋅δₓ ⋅k + 1    1     ╲╱  8⋅δₓ ⋅k + 1    1   ⎬\n",
       "⎪- ──────────────── - ─: 1, ──────────────── - ─: 1⎪\n",
       "⎩         2           2            2           2   ⎭"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fm.eigenvals()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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2H6q7xsQ7/6bde57zrBzwvP2BywWG3BPUTowE15YDJ9pua6pDMICLROpqxqri61bIWmKv\nEE7b3OsuBxhyT1A7MRJaWwGcaDtQdw3BCckPp23VOQfTJ+TQ2grgRFtcwdjPEE7bkZaLq0/IobUV\nwIm28LZrCE5Ifjhtn5pzAvQHObS2AljRdm/sHIMTUOCtrXPNg+qkOk1TUwv0eH+QO9HWixahbMCK\ntmuaOviJTXlr6+WWgprLuNf0B7kTbb1osZINWNF2yPnM4dtBN/WCaVtsud3qD3JgbSWwoi1c5x6E\n8CXBtJ2KM5zvC3JgbSWwqu0ubfYiY0RCFAXT9s4sb/uCHFhbCaxqO1TPGobAZaG0rS5lOd4X5LDa\nJsCqtvwnLbLGIlxZKG3/mO3ygeziIKVhtU2AibbytxYHJoKweBn1/q3F9tbUnx4s3Z5dvx/IbMA9\nf2sx231WagXeR35r8bPytxYj91u+Txdd1YkmyW8tsm1XhvBfFCYnzxXbdp8Y+4HMBrwnqGykrMAn\nTU42/g+5Tzwm2Zo+pgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ \\left ( - \\frac{1}{2} \\sqrt{8 \\delta_{x}^{2} k + 1} - \\frac{1}{2}, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}- \\frac{\\delta_{x} k}{\\frac{1}{2} \\sqrt{8 \\delta_{x}^{2} k + 1} + \\frac{1}{2}}\\\\1\\end{matrix}\\right]\\right ]\\right ), \\quad \\left ( \\frac{1}{2} \\sqrt{8 \\delta_{x}^{2} k + 1} - \\frac{1}{2}, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}- \\frac{\\delta_{x} k}{- \\frac{1}{2} \\sqrt{8 \\delta_{x}^{2} k + 1} + \\frac{1}{2}}\\\\1\\end{matrix}\\right]\\right ]\\right )\\right ]$$"
      ],
      "text/plain": [
       "⎡⎛     _____________                                 ⎞  ⎛   _____________     \n",
       "⎢⎜    ╱     2                                        ⎟  ⎜  ╱     2            \n",
       "⎢⎜  ╲╱  8⋅δₓ ⋅k + 1    1     ⎡⎡       -δₓ⋅k        ⎤⎤⎟  ⎜╲╱  8⋅δₓ ⋅k + 1    1 \n",
       "⎢⎜- ──────────────── - ─, 1, ⎢⎢────────────────────⎥⎥⎟, ⎜──────────────── - ─,\n",
       "⎢⎜         2           2     ⎢⎢   _____________    ⎥⎥⎟  ⎜       2           2 \n",
       "⎢⎜                           ⎢⎢  ╱     2           ⎥⎥⎟  ⎜                     \n",
       "⎢⎜                           ⎢⎢╲╱  8⋅δₓ ⋅k + 1    1⎥⎥⎟  ⎜                     \n",
       "⎢⎜                           ⎢⎢──────────────── + ─⎥⎥⎟  ⎜                     \n",
       "⎢⎜                           ⎢⎢       2           2⎥⎥⎟  ⎜                     \n",
       "⎢⎜                           ⎢⎢                    ⎥⎥⎟  ⎜                     \n",
       "⎣⎝                           ⎣⎣         1          ⎦⎦⎠  ⎝                     \n",
       "\n",
       "                              ⎞⎤\n",
       "                              ⎟⎥\n",
       "    ⎡⎡        -δₓ⋅k         ⎤⎤⎟⎥\n",
       " 1, ⎢⎢──────────────────────⎥⎥⎟⎥\n",
       "    ⎢⎢     _____________    ⎥⎥⎟⎥\n",
       "    ⎢⎢    ╱     2           ⎥⎥⎟⎥\n",
       "    ⎢⎢  ╲╱  8⋅δₓ ⋅k + 1    1⎥⎥⎟⎥\n",
       "    ⎢⎢- ──────────────── + ─⎥⎥⎟⎥\n",
       "    ⎢⎢         2           2⎥⎥⎟⎥\n",
       "    ⎢⎢                      ⎥⎥⎟⎥\n",
       "    ⎣⎣          1           ⎦⎦⎠⎦"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vals = Fm.eigenvects()\n",
    "vals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "(val1,n,[vect1]),(val2,n2,[vect2])=vals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKcAAAAqBAMAAAAkFlbZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSZLlhEbtosgAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- \\frac{1}{2} \\sqrt{8 \\delta_{x}^{2} k + 1} - \\frac{1}{2}$$"
      ],
      "text/plain": [
       "     _____________    \n",
       "    ╱     2           \n",
       "  ╲╱  8⋅δₓ ⋅k + 1    1\n",
       "- ──────────────── - ─\n",
       "         2           2"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "val1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'- \\\\frac{1}{2} \\\\sqrt{8 \\\\delta_{x}^{2} k + 1} - \\\\frac{1}{2}'"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "latex(val1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJEAAABABAMAAADolIiaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARM1UEN0ymSKJdmar\n77t3L27gAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADTklEQVRYCe2YTWgTURDH/0k2az62aaB68GJD\nEClqIRKlp9KlEopSbJQKvTUUpFgsBNqCB8U9CIIIzU0bUXJSarUNCILioUq1UA+KJz+QrngpXqxQ\nFUEa573NZkO2m31akR7yIPPmzZv3y8zsbJYs4D7kiTbuFHpb17e1VIrVdQCCiBoex50cv5RKCbQm\nk2U/JzeEisaWtObk0pLsIpLTrmWXvhu6J23ZarUmIZJvwjgXzI7M1xLMtRhpoclIz3t/u2N+QqRg\nUcnyr25+bkZgn4VIzapS4EdHZxN2RNkiROqDP8r9Z+VXGSeUEMmvDiEwPTOPz3I6tylSYGkcPo+e\nQxuWNhcTDyNUcIrGsAtlx1yDkWixLkqYpDzKZ0RI3SfZ2AOU/m7QdwjHVDcetvm/SJN1cl2tiXKD\nmBYslyeW6qwpKt+zk3a+qRwKxExVnlox1drZc0PjJjsJAxXfbYlwfhwYPqFFsAO4V3WnDFWc4Bcg\npfAAx4CcB170qeZtzBjDz/6MNIYVYmCGnToEBmNjnglaAgoTIjFJaZyPniFCBpDoMUUJsqEzwUkR\npomQgkWk1rNARxroVYGr8l12UmdChDTyI8FcaaSA8PIAFPU9pLNHgH0HeH4627NI8twYM9TtcarP\nlO+Deh2L8Je+Ab8y/IjOpEXiNhK2Lmimtl6FJ0p7aVClUok1LHFv+d0nKnI8/jq+l5Oextvj8QK/\n59m+jYTTNzVqnRhAP20BHaGDqzCe4SF9VGZHdCasmMymtZN0etAGjn6la0IH2qhK/XKMnUWkcCrI\nZp2JTib4teNNSws7KcYe/tI68JC2e/LXcOW2Shrg1ZRbbNbpc2mQZcxJvGlpYSfBRxcdFzT002SO\ny/nHpmrExFdGPxmZbkTq1shtMhvOVc4CiV08MW7RK3aDxJqWhj0mif+qKGt+lTuUxR1roVXUANd6\nVT7ZST0XWTDyzw6+Xxa5j+HqZbXOm5YMdtJgidUJ7bur3V9M82JXm0ydNy0t7KSyR1fWdBWcHUle\nTZBgujmSTAfhuUGiUnWqrvUSqpO0+PIfkYC5BknkH1CjTkKduSX76dzy/qLbjSeYnRuG9hskgSJt\n4Tq5v6Fxz4+/oTns/obGndSSTGq/AbYfFCsoenq9AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{\\delta_{x} k}{\\frac{1}{2} \\sqrt{8 \\delta_{x}^{2} k + 1} + \\frac{1}{2}}\\\\1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡       -δₓ⋅k        ⎤\n",
       "⎢────────────────────⎥\n",
       "⎢   _____________    ⎥\n",
       "⎢  ╱     2           ⎥\n",
       "⎢╲╱  8⋅δₓ ⋅k + 1    1⎥\n",
       "⎢──────────────── + ─⎥\n",
       "⎢       2           2⎥\n",
       "⎢                    ⎥\n",
       "⎣         1          ⎦"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vect1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJYAAAAqBAMAAACjJ7/fAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nONs119zS4ln6zVB57Z5rtlj8jT/a/u30fc8yBgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{1}{2} \\sqrt{8 \\delta_{x}^{2} k + 1} - \\frac{1}{2}$$"
      ],
      "text/plain": [
       "   _____________    \n",
       "  ╱     2           \n",
       "╲╱  8⋅δₓ ⋅k + 1    1\n",
       "──────────────── - ─\n",
       "       2           2"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "val2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'\\\\frac{1}{2} \\\\sqrt{8 \\\\delta_{x}^{2} k + 1} - \\\\frac{1}{2}'"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "latex(val2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ0AAABABAMAAADyqGgUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{\\delta_{x} k}{- \\frac{1}{2} \\sqrt{8 \\delta_{x}^{2} k + 1} + \\frac{1}{2}}\\\\1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡        -δₓ⋅k         ⎤\n",
       "⎢──────────────────────⎥\n",
       "⎢     _____________    ⎥\n",
       "⎢    ╱     2           ⎥\n",
       "⎢  ╲╱  8⋅δₓ ⋅k + 1    1⎥\n",
       "⎢- ──────────────── + ─⎥\n",
       "⎢         2           2⎥\n",
       "⎢                      ⎥\n",
       "⎣          1           ⎦"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vect2"
   ]
  }
 ],
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