Lecture05.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lecture 5\n",
    "\n",
    "This lecture covers linear difference equations (recurrence equations). Below are some examples solved using SymPy.\n",
    "\n",
    "We first import the necessary modules:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()\n",
    "from IPython.display import display\n",
    "\n",
    "from fractions import Fraction\n",
    "\n",
    "# Support for interactive plots\n",
    "from ipywidgets import interact\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Student grant example\n",
    "\n",
    "This example is covered in the lecture (solved by hand). Here it is solved using SymPy.\n",
    "\n",
    "The difference equation (non-homegeneous case) for the student grant example from the lecture notes has the form\n",
    "\n",
    "$$\n",
    "y_{n} - 2.95 y_{n-1} + 2 y_{n-2} = −(63.685)(1.07^{n})\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We first define $n$ as an integer symbol and $y$ as a function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = Symbol(\"n\", integer=True)\n",
    "y = Function(\"y\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we define the difference equation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARMAAAAqBAMAAABihaJaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nqKAy/dMAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$y{\\left (n \\right )} + 2 y{\\left (n - 2 \\right )} - \\frac{59}{20} y{\\left (n - 1 \\right )}$$"
      ],
      "text/plain": [
       "                    59⋅y(n - 1)\n",
       "y(n) + 2⋅y(n - 2) - ───────────\n",
       "                         20    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = y(n) - Fraction(295, 100)*y(n - 1) + 2*y(n - 2)\n",
    "display(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll solve the homogeneous version of the equation using `rsolve`. To compare to the solution in the lecture notes, we'll also evaluate the the solution in floating point:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYcAAABBBAMAAADI9nVQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
      "text/latex": [
       "$$C_{0} \\left(- \\frac{\\sqrt{281}}{40} + \\frac{59}{40}\\right)^{n} + C_{1} \\left(\\frac{\\sqrt{281}}{40} + \\frac{59}{40}\\right)^{n}$$"
      ],
      "text/plain": [
       "                n                 n\n",
       "   ⎛  √281   59⎞       ⎛√281   59⎞ \n",
       "C₀⋅⎜- ──── + ──⎟  + C₁⋅⎜──── + ──⎟ \n",
       "   ⎝   40    40⎠       ⎝ 40    40⎠ "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAaAAAAATBAMAAAAkMcQFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrU6JjD9I4AAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1.05592363464399^{n} C_{0} + 1.89407636535601^{n} C_{1}$$"
      ],
      "text/plain": [
       "                n                      n   \n",
       "1.05592363464399 ⋅C₀ + 1.89407636535601 ⋅C₁"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqn = Eq(f, 0)\n",
    "soln = rsolve(eqn, y(n))\n",
    "display(soln)\n",
    "soln.evalf()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The solution is the same as in the lecture notes.\n",
    "\n",
    "We now consider the non-homogeneous case:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAbcAAAAqBAMAAADCNgjxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$y{\\left (n \\right )} + 2 y{\\left (n - 2 \\right )} - \\frac{59}{20} y{\\left (n - 1 \\right )} = - 63.685 \\cdot 1.07^{n}$$"
      ],
      "text/plain": [
       "                    59⋅y(n - 1)               n\n",
       "y(n) + 2⋅y(n - 2) - ─────────── = -63.685⋅1.07 \n",
       "                         20                    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Create non-homogeneous equation\n",
    "eqn = Eq(f, -63.685*(1.07**n))\n",
    "display(eqn)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solving the non-homogeneous equation, on this occasion providing the initial conditions $y(0) = 2000$ and $y(1) = 2200$,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA3kAAABBBAMAAACeH3C7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXYyEM1Embsi72bd\nVKu+2mc6AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAOS0lEQVR4Ae1bfYxcVRU/s/NmZmdm52NLTEmQ\n7lIlDfjRqf/wj3EfmIakadg1sJgItBMTNSFqpwosQqAbJFqN0iHBENCka2iRD6FLRY3F2MWIH3/Q\njoGEBBIZIKaRanaBrdCWdv2dc+/b92bee+3befNeXxNP0vfOPfecc3/n3Hn33Xfuluj/5MjAb3+1\ntuFoJpg9f5DGl8TfPEmPxzdamJHOH6RBotwfRMlPJ6sfuPxVc7Top9RveU+IzwnSfkfu8leac4lW\nIrhHK5dHszMrsQuh2yPic4A0RJABTR+uBVT0VttUV/JCLVcN58nbv4e0R8TnAKkH+D6L/hjOX1a/\n7XbT0Ddimr0eEZ8DpOFSG8A6NeFWyi91kVvFltyq2G/SwEW2MErOiTjZSKPMgvK9u+4eo+whc2tp\nyXjLtyuaDifiZCONJn6n153OhuZ/4iHzFaVmfLui6XAiTjbSaOJ3eC0ddzQs9i6LCXI33g2i1T+d\nDsSJRtq/mP08ZabdPcYcXX37d4muuuQ2d6dIbjCvaNHVw/+SxojpoxWN2Ik42Uijid/pdWzW2VJ8\nqmW0aaRODSrjnfaq6dbYt/QOGU/Sw9K128OF26RvEifiZCPtW8i+juY9ejZTzqTCTGmWjAnjjXnT\nrfKFN5tUnqFyg7syE26FCCVOxMlGGmESlGvjtMcQF1FhjsrHU9OElYkOmm6VNkSVCUotcNeA16uT\nOyKhDsSJRhpJ+B1Oi//taEoDU5ZZpPJ72Q/MFBZFv9kbn9bzZrzj9hGdxIk42Uijy4HleVCeHqul\n7gOYMqKhRZo/eTEYr9k7uv/bVMCzp+b+S8ounqsTcbKRRp+PynTnGF9Fc0hE26qUWroOrNfs3Uu7\n61g2MydFdaQmt3guFuLkI40+H5gjJw3sRWutSA4RvfL1E3Xv2SManKN76O+nRHXbqNziuWjE5wHS\n6POB7wInPbAe7TmW5BpUbNP6PX6zlztOpf0b3mNVwhswPtKIzwOk0edkq9k5BiaiOMqidUQFk9J4\nuDxWzrJ+5ZXVeezQDFvERBbi5CONPiEfdg2BR2rIhKzUoIfGcT/qOXvYsOR44jJtXAgfF3KL52Ih\nTj7SyPOR/6BriPz79AyLNhBdVmgS/cBz9nLY2kwU99AYNEADavMifNSXZcRJR1q88zNYvyKl0vvd\n7ufNP0GUf3747rnsdbKKrje7dSjfoAPNgeuMG1RPMcYPPhtxwpEWb57ln3+UNOA6Hxi7uIEBMzid\nnaPPbUeV+u59X5x1QXhwzS1Ew1N11ZH1Kti4bPojsBGHRrrqkpqjGi+Fd2C0C/DG9jVIf2rtBaQ4\nKdpL+f6zR4aHq5ln03eRcM7IUp+4gNKHtg8PP/BJqpjOnnSDjKmdNTLWrq6TGKNXJE4txUOXiA1Y\nF3QFcMCxGm55zBx2JZ2U2Wt2CgK08sfcSqueqbEwP3xhS2GwQSvk9w//mBWAS3MMVvKl2+i0DzmW\no7QRh0W6sZ6bcFTjufDOZBfgN83m95BxPd1KwknRXhkcxI+7XVhaqpJwYqeL+TfR5mYR3afpTbpZ\nOqzLUJuu4foGXjv/ITFGj0iohK29Pg5QbqCLl1Nb6YLF5xs7VsMtj1l2FcrslQlGQemESzHTMv7G\nwgepdFJhsEELciy+r/MzfYiXYeEAVrKk2xyGOuTg2FSU0LcRh0X6fRqctqvxxIV3JrsA/zbRERps\n0E4STor2YkDfAQ7KrLmMFAcrq5hfnKHURA6e2vQIbU+LR325vE1YsJ6g14g2KmN0iOSaNYvag+UG\nukS4iC42Fk+TOFbDyejsNOPeLWKf4kelUZ+efS75AaKPsHB9iz5UGGzQEvbAKarMCS7NMVjJkrQl\nDMkXSWwSJTt0IA6HNMuhF+xqfJu9g9r4pwrwWFFGaBvPqXBStBcDqhJdgPkDCceM/rAabFNpMYV1\nrpZfoD+/JD3qkv91m35JNN+8kGizMkaHSKiI2dMe5PuMdYkvokv00y0kjtVwy2MWxIxNg1Bp1Edr\nn9ndcR/R6yy8uY7QBYMNWsLOPso5YlyaY7CSJd1GGJIvkthUlHAYDHEApIMz8OaoxrfRZGrj3/g0\nCvDGB/jxNX+IpuKkaC8GEGVHrQkAJyRpp0qbsqfQ/jJl23TJrOqSawqfV1tNmq9tu5cusoyVxDV7\nrEt8EV38RrYsO5bh9JiVBYf/s7K+OTnc7LbFPmZbS4RYOQWDDVpmD328XgKX5gBWZUm3kQ19yMG/\nTImbVYMhDoC0cu0q7MnsajwX3pnsAvxePHv1YxvW1Uk4XbTn8j0KhVg7n1tXUxyb6WdvvEHZd9Ga\nZlEH/U59HB82y0vPtCxj1jhsumZPdPkiulRsbiHLMQaW0fkWLBesyRQgJ0qRDHyJbBuVxuaqwmCD\n1mEbz+OxAi78tsERg1VZUm3Mns6XWlc4SlAwxAGQjt9AQ/z7sqrxXHjnAewC/Ah+NM8em6UXsYDS\n1lldtGcDwtNDg2aedw3ghBgv0e4qpTF7fLLWScaozF4WW6ORY6ZlDB2WdK2coqsMWJeuoi2WYzWc\nHjNYLiwcvjnZyj9CJ+UBaWyUJff/xVR4bdAq7PxHb1G4SDgBq7Ikbfktq0MOFRtHCQqGOADS8QXK\n8VbvkFWNp8E5HgDTogvwg6OlkZeXiO4wmWvpoj0bkLEgmj/SnImWnr0GpQGUy1SdlFKFKWzNSrcd\nfpT7YMwESffsia5clG6VZ085loH16AFzIaPg4puTXa7Zw7M3VhW7gUcV3g7QgvzyOt7CsnKCE7Aq\nS9ht1VU21CGHmj2OEhRy9myklTnig167Gk+ovQktF+Bf+9hIDYW5g3VirqiK9jDAZmtCNHc0hTPw\nvu5cOS+Vbr5snGS6ll5SszdF9AcaOGGiB8ZMkHTPnujKRXRLNWRJr5wysB6dxmfYgSEjTNaogg8V\nT4JSanLyxr2Tk202sHUWuEm76nLDRYN1rJz0hCkYOkAL8sIewQWrwh4dHWdJtTkbOl9q9jhKkELc\nB6RDE1Q8aVfjy1bh3brLfm7efAHzMotx501dtF/HKIamif7Jr0Xh5HFUz14Fz9Ypor+ykpOMqsze\nQIMID/xYSxlDgyVdsye6ykB0ryTMnnbMA6sxcde5ABeESqM+WvbsWQqnsSlpofEQprYlGGzQgjzf\npMFTgks4HR1nSdryWy6oQw6ZPYkSDoMhDoC0PMPPXsmqxhd04d26qz3GFv6IwbPH+ePV8KgYYAVo\nE7+kd5jCocN69ganZSoQfieVjkzt+MUovUL5Bxaw4ZlVxtCBpNk1e6L7KTb4uOj+Y2rqxLe0Yx54\neUws/iugADmxvO3U4SyZtOtlxvCsDVqQVxYxe4JLOB2d/Mq4R8xVvlRsEiW8B0McAGl6kXIzdjU+\nh5/0BPzLXRXgD5jp4zRm0h0kXKHJRfsNKN/j4WlwKYFeUJwEfdDkW3EGX+uUxovDRYU2pas00Hwc\nb9amMkY1jCVdswdL6MpFdMFaX+sysLpAWlnAJTD55sS1ayHra/0x3hELBhu0IC9Mq63302RxQKyy\npHqQDZUviU1FCaTBEAdB+gZtatnVeFR4DjTxApO7KsBfam6cpXLVeIqEk6K9GGBdqXIZqYiYmBNa\nbxK//16kTTUqqR2W7tE3xPmz4dX/pq+YtF8Zj08oCQ3IMg0P+MLkC+dELqILFmkUx2o4a8xguYC5\nUJCcaNWhlvEIAd1zXCkTDDZoCbs4q+pjj5HFAbFkSbcRhj7k4NgkbvYdDHEQpKntb5GjGi+F90OY\nlOUCfGo1fxCuXlMjxXHRXhnQwy3UQrYfMUk4xiXFfJhfufYtdD3Fok5K7To9u3Vp6UNKH1ldV8bY\niIkk9/axO7UHdSbAusQX0SW6cOmocqyG49GZguVC6Z5hz3m4ZqlYd2PVp2u8Gc5O3V5XeG3QKuyv\nTX0PyoxLcQxWZUnaEoYcckhsEiU7D4bYd/bcSC3E5+Hdt+6UbhDpYwI7LGdOrsAulM8shNy1Ftuo\nz5wLcWKR9jlwD3dS9fGQ82NjHRPY3YZp81gk5MxCJPtsedScC3FikUadCWx+3vMZ43JsIfQxgafG\n8pmF9KIgERe5ECcWafQZsc86O8figv992FCZnWK7tXxmwSIDBYm4qBtxcpFGnxG/vyfi0wl8cepj\nAg8cjjMLfLTgLCUu6kacXKTRZ8TzwwTDouDvLHa5gDjPLLAX9fo4ddn0R9CNOLlI+xPvmbykXSUd\n0eaCv31M4OHAeWaBWh3qhXFRF+IEI40hI1wddxMX/PH3ktYxgVvBcWaBzpzf3sdtGF7SiTjJSMPH\nejYPhz0V+HTiTCun88wCDjILnl6iEXYiTjLSaOJ3et3hqpKgVxX8rWMCp7rmnWcWEFXmPHSiEnUg\nTjTSqDJg+5W/xbObilMFf+uYoLsXbeeZBZpccY+NOhAnGmn0KRlve49ROMvXun1mAXv82V981I04\nuUijz8nQnPcYBRzT8DGBLy2fWUBj3lcrgo5uxMlFGkHwXS5T7j/HZQ0u+MsxQZf6ctM+s4AIMxkf\ndSFOMNLoc5LnPz4MSdkYP/dwdhYCcbxIQ6Y1iPmNQZTOrKP/B+2ZlfrXGwJxzEj7F7Ofp4Nenwx+\nyt7ySpxbTvzBS++IY0bqna5+SivV0N5CpLOXsUMgjhlpL9GtzCaFv9QLSZMh7VdoHgJxzEhXGFgv\n6j/vxchpk42zyskD94w4dqTONEXDq79DDOE7E+9rz/q71x4Qx460B4wrNMmEffG9aq5wxLDqPSOO\nHWnYSM9uj78XDkdPhjNfuXXPiGNHuvLYVmzx+xVbdBiU2x3NOBo9Ij4HSKPPRjnce6vHVIaJq0fE\n5wBpmCgD2t4UUM9TLY+/5Y+dekJ8TpBGn5rPhxki3wxj3aNtT4g7kf4PorsF+XLlHSMAAAAASUVO\nRK5CYII=\n",
      "text/latex": [
       "$$6285.5996982757 \\cdot 1.07^{n} - 4285.21114484077 \\left(- \\frac{\\sqrt{281}}{40} + \\frac{59}{40}\\right)^{n} - 0.388553434931812 \\left(\\frac{\\sqrt{281}}{40} + \\frac{59}{40}\\right)^{n}$$"
      ],
      "text/plain": [
       "                                                      n                       \n",
       "                    n                    ⎛  √281   59⎞                      ⎛√\n",
       "6285.5996982757⋅1.07  - 4285.21114484077⋅⎜- ──── + ──⎟  - 0.388553434931812⋅⎜─\n",
       "                                         ⎝   40    40⎠                      ⎝ \n",
       "\n",
       "         n\n",
       "281   59⎞ \n",
       "─── + ──⎟ \n",
       "40    40⎠ "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA6wAAAARBAMAAAArqTxBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
      "text/latex": [
       "$$- 4285.21114484077 \\cdot 1.05592363464399^{n} + 6285.5996982757 \\cdot 1.07^{n} - 0.388553434931812 \\cdot 1.89407636535601^{n}$$"
      ],
      "text/plain": [
       "                                   n                       n                  \n",
       "- 4285.21114484077⋅1.05592363464399  + 6285.5996982757⋅1.07  - 0.3885534349318\n",
       "\n",
       "                   n\n",
       "12⋅1.89407636535601 "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "soln = rsolve(eqn, y(n), init={y(0) : 2000, y(1) : 2200})\n",
    "display(soln)\n",
    "soln.evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZ8AAADzCAYAAABZoxsBAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nHxERCTiFj4iIBJzCR0REAk7hIyIiAff/bZVa3eHneWAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f0903830a20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Fee after 10 years (n=10): 4749.72910622520\n"
     ]
    }
   ],
   "source": [
    "# Plot\n",
    "plot(soln, (n, 0, 12), xlabel=\"year (n)\", ylabel=\"fee\")\n",
    "\n",
    "# Evaluate fee at 10 years\n",
    "print(\"Fee after 10 years (n=10): {}\".format(soln.subs(n, 10).evalf()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## General case\n",
    "\n",
    "Below is the general case for a constant coefficient, second-order difference equation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQsAAAAUBAMAAABojAWbAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nheKAbH8BGs7x0GCe5DYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$a y{\\left (n \\right )} - b y{\\left (n - 1 \\right )} + c y{\\left (n - 2 \\right )}$$"
      ],
      "text/plain": [
       "a⋅y(n) - b⋅y(n - 1) + c⋅y(n - 2)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = Symbol(\"n\", integer=True)\n",
    "y = Function(\"y\")\n",
    "\n",
    "a, b, c = symbols(\"a b c\")\n",
    "\n",
    "f = a*y(n) - b*y(n - 1) + c*y(n - 2)\n",
    "display(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solving for the homogeneous case,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAjEAAAA0BAMAAACJE3RdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\njnKIwW68b1tJpoEEu6oa0/8ACoYFW6y3sB4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$C_{0} \\left(\\frac{1}{2 a} \\left(b - \\sqrt{- 4 a c + b^{2}}\\right)\\right)^{n} + C_{1} \\left(\\frac{1}{2 a} \\left(b + \\sqrt{- 4 a c + b^{2}}\\right)\\right)^{n}$$"
      ],
      "text/plain": [
       "                         n                            n\n",
       "   ⎛       _____________⎞       ⎛       _____________⎞ \n",
       "   ⎜      ╱           2 ⎟       ⎜      ╱           2 ⎟ \n",
       "   ⎜b - ╲╱  -4⋅a⋅c + b  ⎟       ⎜b + ╲╱  -4⋅a⋅c + b  ⎟ \n",
       "C₀⋅⎜────────────────────⎟  + C₁⋅⎜────────────────────⎟ \n",
       "   ⎝        2⋅a         ⎠       ⎝        2⋅a         ⎠ "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eqn = Eq(f, 0)\n",
    "soln = rsolve(eqn, y(n))\n",
    "display(soln)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The above is a general solution. Solving for some initial conditions,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA3kAAAA3BAMAAABDZJ5nAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{\\left(\\frac{1}{2 a} \\left(b - \\sqrt{- 4 a c + b^{2}}\\right)\\right)^{n}}{\\sqrt{- 4 a c + b^{2}}} \\left(- a + b + \\sqrt{- 4 a c + b^{2}}\\right) + \\frac{\\left(\\frac{1}{2 a} \\left(b + \\sqrt{- 4 a c + b^{2}}\\right)\\right)^{n}}{\\sqrt{- 4 a c + b^{2}}} \\left(a - b + \\sqrt{- 4 a c + b^{2}}\\right)$$"
      ],
      "text/plain": [
       "                      n                                                     n \n",
       "⎛       _____________⎞                                ⎛       _____________⎞  \n",
       "⎜      ╱           2 ⎟  ⎛            _____________⎞   ⎜      ╱           2 ⎟  \n",
       "⎜b - ╲╱  -4⋅a⋅c + b  ⎟  ⎜           ╱           2 ⎟   ⎜b + ╲╱  -4⋅a⋅c + b  ⎟  \n",
       "⎜────────────────────⎟ ⋅⎝-a + b + ╲╱  -4⋅a⋅c + b  ⎠   ⎜────────────────────⎟ ⋅\n",
       "⎝        2⋅a         ⎠                                ⎝        2⋅a         ⎠  \n",
       "─────────────────────────────────────────────────── + ────────────────────────\n",
       "                     _____________                                        ____\n",
       "                    ╱           2                                        ╱    \n",
       "                  ╲╱  -4⋅a⋅c + b                                       ╲╱  -4⋅\n",
       "\n",
       "                          \n",
       "                          \n",
       "⎛           _____________⎞\n",
       "⎜          ╱           2 ⎟\n",
       "⎝a - b + ╲╱  -4⋅a⋅c + b  ⎠\n",
       "                          \n",
       "──────────────────────────\n",
       "_________                 \n",
       "       2                  \n",
       "a⋅c + b                   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "soln = rsolve(eqn, y(n), init={y(0) : 2, y(1) : 1})\n",
    "display(soln)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we now specify the values for the constants, the solution can be plotted:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAkMAAABBBAMAAAAgQL2KAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYv8DvpNtyS4vpnIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left(\\frac{1}{8} - \\frac{\\sqrt{31} i}{8}\\right)^{n} \\left(\\frac{1}{2} + \\frac{15 i}{62} \\sqrt{31}\\right) + \\left(\\frac{1}{8} + \\frac{\\sqrt{31} i}{8}\\right)^{n} \\left(\\frac{1}{2} - \\frac{15 i}{62} \\sqrt{31}\\right)$$"
      ],
      "text/plain": [
       "           n                             n               \n",
       "⎛1   √31⋅ⅈ⎞  ⎛1   15⋅√31⋅ⅈ⎞   ⎛1   √31⋅ⅈ⎞  ⎛1   15⋅√31⋅ⅈ⎞\n",
       "⎜─ - ─────⎟ ⋅⎜─ + ────────⎟ + ⎜─ + ─────⎟ ⋅⎜─ - ────────⎟\n",
       "⎝8     8  ⎠  ⎝2      62   ⎠   ⎝8     8  ⎠  ⎝2      62   ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZIAAADxCAYAAADhlTG6AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMS4wLCBo\ndHRwOi8vbWF0cGxvdGxpYi5vcmcvpW3flQAAIABJREFUeJzt3Xl8VfWd//HX92bfb/Y9kECABLIA\nYVEURMRitTJFrXu1LmiXqdrOdOxv7DIdpzJd7Njq2Fp01KF1aTuKW1XADVSWyE7CmmD2jez7Xb6/\nP3KxSANJyL055977eT4e90HuzVk+F0je95zvprTWCCGEEOfKYnQBQgghvJsEiRBCiHGRIBFCCDEu\nEiRCCCHGRYJECCHEuEiQCCGEGBcJEiGEEOMiQSKEEGJcAseysVLqecDmelqvtf6e+0sSQgjhTcYU\nJMDHWutHAJRS8R6oRwghhJcZa5CsVEp1A5u11oc9UZAQQgjvosYy15ZSKg0ocj2maK3v9FRhQggh\nvMOIQaKU2gD8k9Z6z8SU5DYyG6UQQoydGvMOowiSOcAvgePA/9Na159TaRNPgkQIIcZuzEEyYvdf\nrfVOrfVS4DXgTaXUj5RSYedSnRBCCN8zqnEkSikFHAIeB/4ROKKUutmThQkhhPAOo7m19SGQDRwA\ntgLbgIPAPUCI1nq1p4s8R3JrSwghxs4jbSQzgTI9zIZKqXKtdd5YTzpBJEiEEGLs3B8kZ91ZqRyt\ndcU5H8CzJEiEEGLsJjZITM5n35gQQniQ+3ttCXA6nfhw4AohxLiMdYoUv3KksZOXdtZy/EQPgRYL\nucmRXF2SSWqM9H4WQoiTfPaKZMWKFePa/+OKE/zktXLKGzqZFB9OYnQw2yvb+PH6A5TVdbipSiGE\n8H4+e0XS0tJyzvuW1XXyXxsOkRAZwg8uzyfFOnQFcqShiwffKOfHrxzgF9cUkxUf7q5yhRDCa/ns\nFcm5GrQ7+c2mw7T12PjBFX8LEYDclCh+9KV8jrX08Iu3D2FzOA2sVAghzEGC5DQvllZztLmH762Y\nTsowbSE5iZE8uHIWr+yp40+lNQZUKIQQ5iJBcorOfhsbyxrJTojgkvyUM253WUEq18zN4KdvlNPS\n3T+BFQohhPlIkJxi/a5aGjv7+ceLc0fc9q4lU8hJiOCxd49NQGVCCGFeEiQudoeTx987RlxkMAUZ\nMSNuPzUpkgU5cZQeb6O5a2ACKhRCCHOSIHH54EgzDZ393HLe5FHvc/38LPbXdfBiaZXnChNCCJMz\nPEiUUplKqXeVUmVKqQNKqXuG2UYppX6tlDqqlNrrWmzLrdZtrWJ6chTL8pJHvU9OYiRfmZvJ2s2V\n9Nsc7i5JCCG8guFBAtiB72qt84GFwDeVUvmnbXMZkOt6rGZoXRS3aeke4P3DzSyZnkSAZWzTzFxZ\nnEbfoINN5U3uLEkIIbyG4UGita7XWu90fd0FlAPpp222EnhWD9kKWJVSqe6qYWN5I06tWTXn9NOO\n7Pwp8WQnRvDsx8fdVY4QQngVw4PkVEqpycBshhbPOlU6UH3K8xr+PmzO2Us7a5mbFcu05Kgx76uU\n4orCNLZVtlJ1otddJQkhhNcwTZAopSKBvwD3aq07z/EYq5VSpUqp0ubm5lHt09ozyI7jrZw3Jf5c\nTgnAl2enoxT8eacMUBRC+B9TBIlSKoihEPmD1vr/htmkFsg85XmG67XP0Vo/obUu0VqXJCYmjurc\n7xxswqnh0rMMQBxJmjWMq2ZnsL+2Q6abF0L4HcODRCmlgCeBcq31w2fY7BXgq67eWwuBDq11vTvO\nv7GsgdSYUGalR4/rOHMnx/LOwSbK67vcUZYQQngNw4MEWATcDFyslNrtenxRKXW3Uupu1zZvABXA\nUeD3wDfccWKbw8ne2g7XrakxLwr2Ocvzk1EK3jrQ4I7ShBDCaxg+jbzWegsjLO2oh+4XfdPd595d\n3U5dez+FoxjJPpKEyBDmTY7jrQMN3Ld8mhuqE0II72CGKxLDbDnSglJwXk6CW473hZkpHGzoorKl\nxy3HE0IIb+DXQfLh0RYK02OICQ9yy/G+MHNoVLzc3hJC+BO/DZKufhvHT/RwQa57rkYAMmLDKUiP\nkSARQvgVvw2SXVXttHQPsjDn3MePDGfFrBR2VbXT0CHrlAgh/IPfBknp8VYCLIrZWbFuPe7yvGRm\npkXz/mGZe0sI4R/8Nkh2HG8jPzWayBD3dlzLTY7kRPcgHxxucetxhRDCrPwySGwOJ7uq2yiZ7N6r\nERiae+uC3AQ+PNaCwymj3IUQvs8vg2R/bQf9NifzJsd55PgXTE2gvdfGgboOjxxfCCHMxC+DZG9N\nB1MSI5idafXI8RdNHeoJtuWo3N4SQvg+vw2Szn47qdYwjxw/MSqEGSlRbDkiQSKE8H1+GSQH6jqY\nlTa+SRpHcmFuAqXH2+gblCV4hRC+ze+CpN/m4EhTNwXp459f62wuyE1k0OFk+/FWj55HCCGM5ndB\nUl7ficOpmenhIJk/OY7gAAtbjoxugS0hhPBWfhck++uGFl+c5eEgCQsOoGRyLJulnUQI4eP8L0hq\nOogNDyItJtTj51o0NYGDDV00dw14/FxCCGEU/wuSug5mpceMeyGr0bggN4GpSZFsrzzh8XMJIYRR\n/CpIBuwODjd2efy21kn5qdHUtvWx43jbhJxPCCGM4FdBcqSxG5tDMyttYoIkKMDCnElWtldKzy0h\nhO/yqyDZVzs0ZcmsdM+OITlVyaQ4DjZ00tlvm7BzCiHERPKrINlf20FUaCBZceETds55k+Nw6qH1\nT4QQwhf5V5DUdTIrbWIa2k+anWUlwKLYIbe3hBA+ym+CxOZwUl7fOaG3tQAiQgKZmRbNDhnhLoTw\nUaYIEqXUU0qpJqXU/jN8/yKlVIdSarfr8cOxnuNYczc2h3PCemydqmRSHLur2xm0Oyf83EII4Wmm\nCBLgaWDFCNts1loXux4/GesJDjV0ERygmJYcdU4Fjsf87FgG7M7PGvuFEMKXmCJItNYfAB6991PR\n3MOgQ5OdEOHJ0wxr7qShBbRK5faWEMIHmSJIRuk8pdQepdRflVIzh9tAKbVaKVWqlCptbv78ZImV\nLT2kW8MIDQqYkGJPlRgVQnZChAxMFEL4JG8Jkp3AJK11EfAb4OXhNtJaP6G1LtFalyQmJn7uexUt\n3eQkRnq+0jMomRTLJ5+24pR13IUQPsYrgkRr3am17nZ9/QYQpJRKGMP+VDb3kGPAba2T5mXH0dZr\n41hzt2E1CCGEJ3hFkCilUpRr8IdSaj5DdY96JsSmrgF6Bh3kJBoXJCWTYkmOCpEGdyGEzwk0ugAA\npdRzwEVAglKqBvgREASgtf4tcDXwdaWUHegDrtNaj/oe0cmrgJwE425tTY4PZ9DhZGvFCVbNyTCs\nDiGEcDdTBInW+voRvv8o8Oi5Hr+ypQeAbAOvSCwWC0WZVvZUyxWJEMK3eMWtrfGqaO4hNMhCarTn\nF7M6m6IMK0eauugZsBtahxBCuJNfBEllSw+T4yOwWCZujq3hFGdaceqhySOFEMJX+EWQVDR3M8XA\nrr8nFWYMTc+yp0ZmAhZC+A6fD5JBu5Pqtj5De2ydFB8ZQkZsmLSTCCF8is8HSVVrLw6nMVOjDKco\n08ruarkiEUL4Dp8PkoqTXX9NcGsLoDjDSm17Hy3dA0aXIoQQbuHzQfJZ11+TXJGcbCfZK+0kQggf\n4fNBUtHcQ0JkMDFhQUaXAsCs9BgsCnZLO4kQwkf4fpC0dBs6ov10ESGBTEuOYo+0kwghfITPB0ll\nS49pbmudVJRhZU9NO2OY5UUIIUzLp4Oko89GS/egKbr+nqoo00p7r43q1j6jSxFCiHHz6SAxW0P7\nSScb3HdLg7sQwgf4dJCYrevvSdNToggJtEg7iRDCJ/h0kFS29BBgUWTFhRtdyucEBViYlR4jQSKE\n8Ak+HSQVzT1kxoYRHGi+t1mUYWV/XQd2h9PoUoQQYlzM9xvWjY41G7tO+9kUZcbQb3NyuFGW3hVC\neDefDpLjJ8zX9fekogwrIDMBCyG8n88Gic3hpN/mNF3X35MmxYcTExYk7SRCCK/ns0EyYB9qezDr\nFYlSSmYCFkL4BJ8PEjMsaHUmxRkxHG7sondQlt4VQngv3w0Sm4OI4ACSokKMLuWMilxL7x6o6zS6\nFCGEOGe+GyR2J9mJEShl7DrtZ1N4ssFdbm8JIbyYKYJEKfWUUqpJKbX/DN9XSqlfK6WOKqX2KqXm\njHTMAbvTVLP+DicxKoR0a5i0kwghvJopggR4Glhxlu9fBuS6HquBx0c6oM3hNG1D+6mKMmOkC7AQ\nwquZIki01h8ArWfZZCXwrB6yFbAqpVJHOq5Zu/6eqijDSnVrHydk6V0hhJcyRZCMQjpQfcrzGtdr\nn6OUWq2UKlVKlUYHOJiTFTthBZ6roswY8lKjKKuXBnchhHfyliAZFa31E1rrEq11yaSUODJNNlnj\ncGalWznY0MXOT+X2lhDCO3lLkNQCmac8z3C95vUiQwLJTohgX62s4S6E8E7eEiSvAF919d5aCHRo\nreuNLspdCtJj2C9BIoTwUoFGFwCglHoOuAhIUErVAD8CggC01r8F3gC+CBwFeoGvGVOpZxSkx7B+\ndx3NXQMkmngApRBCDMcUQaK1vn6E72vgmxNUzoQrSB9aend/bQdLZyQZXI0QQoyNt9za8mkz02NQ\nCmknEUJ4JQkSEzjZ4L63RoJECOF9JEhMQhrchRDeSoLEJArSY2jo7Ke5S0a4CyG8iwSJSZza4C6E\nEN5EgsQkpMFdCOGtJEhMQka4CyG8lQSJiUiDuxDCG0mQmEhBegz1HdLgLoTwLhIkJiIN7kIIbyRB\nYiLS4C6E8EYSJCYiDe5CCG8kQWIy0uAuhPA2EiQmIw3uQghvI0FiMgXp0UxPjuJgg1yVCCG8gwSJ\nycxMt3K4qYtdVRIkQgjvIEFiMpEhgeTIlPJCCC8iQWJChRlW9tW2G12GEEKMigSJCRWkx9DYOUBT\nZ7/RpQghxIgkSEyoMGNohLuMJxFCeAMJEhPKT4vGopB2EiGEV5AgMaHw4ECmJkXKFYkQwiuYIkiU\nUiuUUoeUUkeVUvcP8/1blVLNSqndrscdRtQ5kQrSreyt6UBrbXQpQghxVoYHiVIqAHgMuAzIB65X\nSuUPs+kLWuti12PthBZpgMKMGFq6B2jslBHuQghzMzxIgPnAUa11hdZ6EHgeWGlwTYYrcDW4762R\nbsBCCHMzQ5CkA9WnPK9xvXa6q5RSe5VSf1ZKZQ53IKXUaqVUqVKqtLm52RO1Tpj81GgCLEraSYQQ\npmeGIBmNV4HJWutCYAPwzHAbaa2f0FqXaK1LEhMTJ7RAdwsNCiA3KVJ6bgkhTC/Q6AKAWuDUK4wM\n12uf0VqfOOXpWuBnE1CX4QozYthY3oTWGqWU0eX4vOrWXt473MzRpi5sDs2JrgGauwfITY5icnw4\nuUlRLMyJIzI0yOhShTAVMwTJDiBXKZXNUIBcB9xw6gZKqVStdb3r6ZVA+cSWaIyCDCsvltZQ19FP\nujXM6HJ8ktaajeVNbCxr5OOKFqpa+5ibFUtmXBhBgRbCggLYW9NORXM3//nmIaYmRTI9JYrVF2ZT\nlBlrdPlCmILhQaK1tiulvgW8BQQAT2mtDyilfgKUaq1fAb6tlLoSsAOtwK2GFTyBCl1ruO+raZcg\n8YBdVW08+Ho5n3zaxsy0aG5ckMWyvGSmJEb+3RXggN3B9opWNpQ3sqe6nZWPfcSFuQl899LpFGda\nDXoHQpiD8tVxCiUlJbq0tNToMsal3+Zg1o/eYvXiHL63YobR5fiMQbuTRzYd5vH3jrFkWiJfLEjl\ny7PTCQwYXZNhv83Buq2f8t/vHaO1Z5A7L8zmjgtzSI4O9XDlQkyIMd9HN/yKRJxZaFAA01OipOeW\nG7X2DLLmr+W8WFrDtSWZ/OBL+USGjO3HIDQogDsuzOG6+Vk8uaWCJ7dU8mJpDT9ZOZMri9KkPUv4\nHQkSkyvMiOGNfQ3S4O4Gte193PzkNmrb+njshtlcXpg2ruNFhgRyz7JpfKkwje/+aQ/3PL+b/bUd\n3HvJNCLGGE5CeDNv6f7rtwrSrXT02ahp6zO6FK9WdaKHqx//iObOAf739gXjDpFT5SRG8ue7z+eB\ny/N4ckslq/77I4639Ljt+EKYnQSJyRV+NsJdbm+dq6bOfq57Yiu5SZG8cNd5zM+Oc/s5AiyKOy7M\n4Znb5tPY1c/NT25jW8WJkXcUwgdIkJjctOQo5k+OpVI+4Z6T7gE7X3t6B+19Nr63Ygb5adEePd+F\nuYm8+q0LmJIYyc1PbWdDWaNHzyeEGUiQmFxwoIVBh+aDI9495YsR7A4n9zy3i4MNXTx24xxmubpT\ne1pmXDi/uraYvJQo7l73CS/tqpmQ8wphFAkSL1CcaWVfTQd2h9PoUrzKI5uO0NjVz09WzmTp9KQJ\nPXdsRDB/uHMh8ybHct8Le/jfj49P6PmFmEgSJF5gdpaVPpuDw43dRpfiNT461sKj7x5lRko0Ny6Y\nZEgNkSGBPP21+VySl8QP1h/gsXePyvoywidJkHiBkyOnd1fLlPKjcaJ7gPte2E12QgT/duVMQ2sJ\nDQrg8Zvm8g/Fafz8rUOs+etBCRPhcyRIvEBWXDhxEcHsqmozuhTT01rz/17aR1uPjd9cP9sU4zmC\nAiw8/JViblqYxYul1fzi7UMSJsKnGP9TJkaklKIoI0auSEbhLztrOdzYzQNX5DEzbWIa10fDYlH8\n+8pZxIYH85t3jjJgc/Kvl+fJIFPhEyRIvERxZizvHW6mq99GlExjPqyW7gEefL2MqYmR3GRQu8jZ\nKKX4zvJpdPXbWbulktCgAP7pC9ONLkuIcZMg8RLFWVa0HhqYuGhqgtHlmNKDr5XRM2DnoVUFWCzm\n/KSvlOJHX8pnwO7g0XePEhxo4dvLco0uS4hxkTYSL1GcIQ3uZ/P+4WZe3l3H1y+aSm5ylNHlnJVS\niv/4hwJWzU7n4Q2HWbf1U6NLEmJc5IrES8SEB5GTEMGuKgmS0w3anfx64xGW5SXxjYumGF3OqFgs\nip9dXUhsRDAPvLwfp9Z89bzJRpclxDmRKxIvUpxpZXd1u/T4Oc2zHx/nk6o2bl44idCgAKPLGbXA\nAAv3XzaDS/KS+eH6A/yptNrokoQ4JxIkXmR2lpWW7gFq22Um4JNaewZ5ZNMRlkxL5KIJHr3uDkEB\nFh69YTYX5ibwL3/Zy6t76owuSYgxkyDxIsWuNcKlneRvHtl4mN5BB/96eZ7RpZyz0KAAfnfzXOZO\niuW+F3azUSZ6FF5G2ki8yIzUKEICLeyuaucKN66n4a2ONXWxblsV18/PZJrJG9hHEh4cyFO3zuOm\ntdv4wfr9BAcqFk/zvissb6K1pnvAzonuQU70DNA74KBn0EGARRFgGeoUERJgISw4gITIEBKjQrzq\n1ulEkiDxIkEBFmaly8DEkx7ecJh5k2K575JpRpfiFlGhQTxz23zueX4Xdz77CU/dOk+6ertBW88g\nZfWdHKjroKyuE7tT88mnbZzoHmTwlIlQZ6ZGc6C+83P7pkSH0NA58NnzyJBA4iODyYgNIzEyhCmJ\nkUxNimRKUiST4yMIDvTPmzwSJF6mONPKuq2fYnM4CQrwz/+0AIcbu3hjfwPfvGgq8ZEhRpfjNtbw\nYB7+SjE3/H4btz+zg6duncf5UyRMxqKpq593DzZxqKGbN/fXU9fR/9n3UmNCuSA3gUVTE4iPDCY+\nIpj4iBDiIoOJCw8iMMCC0wlOrXFojd2u6R600dI1SHP3AC3dA7R0D9I3aGfH8TZe3v23Nq0Ai2JS\nXDhFmVYy48IpTI+hMCOGpOhQI/4aJpQEiZcpzrTy5JZKDjV0Tdj6Gmb0601HCA8K4PYLso0uxe3i\nI0P4w50LuP6Jrdz+dCn/87V5LMyJN7os09JaU17fxabyRjYebGKP64p9SmIE87LjmJkWTX5qDPlp\n0cRFBLv13D0Ddipbejja1M3Rpm6ONXfT2Wfj0XeO4HR1rkyODqEgPYaCdCvFWTEUZVixhru3DqOZ\nIkiUUiuAR4AAYK3Wes1p3w8BngXmAieAa7XWxye6TjM4ORPwrup2vw2SI41dvL6vnq8vmUKsm38x\nmEVCZAh/vHMh1/9+Kz959QAPXJ7P+XKb6zNaa3ZWtfH2gUZe3VP32VVHUaaV7y6fxrK8ZPJSozw+\nl1lESCCz0mP+7mexd9BOWV0ne2s62Ffbwd6adko/baN9ow2AnMQIZmfGMj87lplpMcxIiSLQi+8w\nKKPHJCilAoDDwHKgBtgBXK+1Ljtlm28AhVrru5VS1wFf1lpfe7bjlpSU6NLSUg9WbgytNfP+YyNL\npiXxy68UGV2OIb793C42ljey5V8udvsnTLNp7OzjG3/Yxb7aDv77hjlckp9sdEmG6uiz8dLOGp7b\nXs2hxi4WZMcRHRbEJXlJLJ2RRFKUeW8jdfbZ2F/Xwa6qdtejjZzECHYcbyM0yEJhupXiLCvFmVaK\nMqykWUONmtRzzCc1wxXJfOCo1roCQCn1PLASKDtlm5XAj11f/xl4VCmltNEpaACllGtgon9OKX+0\nqZtX99Zx1+IpPh8iAMnRYaz9agm3/s927lr3Cb+8poh/mJ1udFkTaujqo53ntlfx2t46+m1OCjNi\nWLOqgC8VpZliqYDRiA4L4vwpCZ+1eWmtqWrtZXd1O7urh8Ll6Q+Po5TG5tDERQQP3RLLsFKcGcO0\n5CjSrWGmnDHaDFckVwMrtNZ3uJ7fDCzQWn/rlG32u7apcT0/5tqm5bRjrQZWAwQFBc0tLCycoHcx\nsZq6Bmjs7Cc/NZoAk05OOF7Nzc0kJib+3evVrb109tuZnhJFoBe/9zO9vzNxas3xll56Bu2kWcOI\nN3mIjvX9DUcDHb2DdPbb6eizYVEKa3gQcRHBhBnYDdcd7+1MtIZ+m4Nem4O+waFHv91BRHAgPYN2\nLEoRFmQhLDiQ4EALIYEWggMtBLvxttgnn3xyQGs9ayz7eEeUj5LW+gngCYCIiAifvLUFsK3yBNf9\neiMP37WUpTN8c6xBSUkJp//7VTR3c8nD73PvhTl8/4veOwARhn9/I+m3OfjWH3eysbyJu76Yxx0X\nZpvy0ymc2/s7qXvAzvPbq3hySyX2jn5mJ0fy1fMns7I4nUgTXH2M572di95BO4cauiir7+RgfRfl\nri7KpZ+20Q/0A8GBFrLjIyjIiCE0yEJKdCjJrkdKTCiJkSFEhwWN6oOnUqp/xI1OY/y/CtQCmac8\nz3C9Ntw2NUqpQCCGoUZ3v1SYbkUHhbKtstVng2Q4j75zlJDAAO5cnGN0KYY4uWzvLzcc5qd/LWd3\nTTs/v7qQ8GAz/BiPX3PXAH/ZWcNj7x6lq9/Oguw4fvrlAi6anmjawJwI4cGBzM6KZXZW7Gevaa1p\n7hqgoqWHStejormHqtZeDjd20d5r+9wxijJj2FPdQURwAJGhgUSFBpEeE8aA4+QATAuBFnXOdzjM\n8D9wB5CrlMpmKDCuA244bZtXgFuAj4GrgXf8sX3kpLDgAAI769hW6T+9eCpbenh5dy23X5BNgg+N\nGxmroAAL//KF6VjDgvjPNw9yrKmb33+1hMy4cKNLO2e17X088f4xnt9RTXxEMBdOTeCuJVMocvVQ\nFH9PKUVSdChJ0aHDdg3vtzlo7OynsXOAhs5+egZsLJ2eRHe/na5+O10DNoICLNR3OLDZnDicDhxO\njd15br9WDQ8SrbVdKfUt4C2Guv8+pbU+oJT6CVCqtX4FeBL4X6XUUaCVobA5q4QE3/4lO39yLNtr\nOugdtPvMJ9JTrV69+nPPH31naBGo1Yu9Y5r4kZz+/sZCKcXdS6YwIyWKbz+3iysf3cJjN8wxVffg\n0by/iuZuHn/vGC/tGroBsWpOOl+/aCrZCRGeLm9cxvNvN1FCgwKYFB/BpPix/12qe4eaB8a0j69+\nsPfV7r8nvXeoiVv/Zwfrbl/ABbnm+QXiCdWtPdz2dCmLpyXygyvyjS7HVI639HDns6VUtPTwwBfz\nuOX8yaZdHfKkvTXtrN1SwTvlzdgcTq6fn8Wdi3NIt4YZXZoY4pXdf8U5KJkcR4BFsa3yhM8HyZNb\njlPZ0s2zt803uhTTmZwQwUvfXMQPXt7PE5sreG1fPQ+tKjDdJJZaaz440sLv3j/GR8dOEBUSyLcu\nnspVczJIjPLfW5W+QoLES0WGBDIrLZptFa1Gl+JRHb02XiytZmVxBqnyiXVYkSGBPPyVIv5vZy0P\nvl7G5b/ezNeXTOGbF08lJNDY2WoH7Q7e2NfA7z6ooLy+k+ToEL5/2QxuWJBFVGiQobUJ9/HeMfln\noZRasX//fqZOncqaNWtG3sGLVFdXs3TpUvLz8znw/it8cvwE/TaH0WW5lcPhYPbs2VxxxRX8cXsV\nvYMO7rjQN+bUam9v5+qrr2bGjBnk5eXx8ccfu+W4SimumpvBxu8s4fKCVH79zlEue2QzHx5tdsvx\nR+tXv/oVM2fOJL/kAhbdvYZFa97h95srGLQ7+NnVhXzwvaXctWSK14TIbbfdRlJSErNm/W1YRWtr\nK8uXLyc3N5fly5fT1ua9g4OHe39KqZ8rpQ4qpfYqpV5SSo3Y68HngsQ15cpjubm5lJWV8dxzz1FW\nVjbift4iMDCQX/7yl5SVlfGr+7+BA8X6zbuNLsutHnnkEfLy8nAqC09/VMkFUxPIS402uiy3uOee\ne1ixYgUHDx5kz5495OW5dzxMfGQI/3XdbJ65bT52h+bHr5Zx9eMf8eb+ehzn2CNntD6tquGRP21k\n/nefon/596mNmYVVd/EvK2aw4b4lfKUk0/ArpLG69dZbefPNNz/32po1a1i2bBlHjhxh2bJlXv1h\ndbj3B2wAZmmtCxmavur7Ix3H54IE15QrISEhBAcHc91117F+/Xqja3Kb1NRU5syZA8DivHTQmg+P\nTOynTk+qqanh9ddf54477qAzegqNnQM+czXS0dHBBx98wO233w5AcHAwVqtnurgumZbIhvsWc+P8\nLBq7+rl73U6W/uI9nv6wkp4Bu9vOM2B38M7BRv75T3v40lP70dOXs7e2g7sX5zDt8Dr+eUEki6cl\nmr4DwJksXryYuLi4z722fv1w+qccAAANfklEQVR6brnlFgBuueUWXn75ZSNKc4vh3p/W+m2t9cn/\nJFsZGtt3Vr7YRpIOVJ98kpGRwbZt2wwsx3PammrR7TU06gKjS3Gbe++9l5/97Gd0dnbSklBEblIk\nS6Z5ZjqKiVZZWUliYiJf+9rX2LNnD3PnzuWRRx4hIsIz3V1DggK4dVE2N583mbcPNLB2SyU/frWM\n1/bWExESyPlT4lk0NYH81OhR/6Lvtzk44Jp48HBjF3/d10DXgJ2o0EAuyUtBh1ax9qfX8POQYC69\n9FIuvfRSj7w3IzU2NpKamgpASkoKjY0+vTTybcALI23ki0HiF7q7u7nqqqtYfNMP2VHbyaDd6fWr\ns7322mskJSUxd+5cHv/LJvrDEk09DchY2e12du7cyW9+8xsWLFjAPffcw5o1a/j3f/93j543wKK4\nrCCVywpS2VnVxsayRt4ua+Shvx4EIC48iJzESOIjg4kNDyY2IpjY8CCcGtp6B2npGqSle4CgAMX7\nh5uxOYZukWXHh7OyOI1l+cksmpJAT1cHV131HSqPHcVqtXLNNdewbt06brrpJo++PyMppXzm/+fp\nlFL/CtiBP4y0rS8GyeemXKmpqSE93bdmS7XZbFx11VXceOON5C+fz+Z1O9lX287cSXEj72xiH374\nIa+88gpvvPEGjkWrUfGTWP/IA1y77lmjS3OLjIwMMjIyWLBgAQBXX331hN9fn5MVy5ysWL63YgaN\nnf18fOwEB+u72FXdxvGWXnb2ttPWM4jdqUmKCqGtd/Cz9cpnpERzx4U5zM4cmu789Cnb12/cSHZ2\n9mcTGq5atYqPPvrI54IkOTmZ+vp6UlNTqa+vJynJ96YpUkrdClwBLBvNLCK+GCQ7gNyBgQEGBwd5\n/vnn+eMf/2h0TW6jteb2228nLy+P73znO5zoHlpPeltlq9cHyUMPPcRDDz3E0aYuLnn4A5Iat/NH\nHwkRGLoNkpmZyaFDh5g+fTqbNm0iP9+4AZbJ0aFDU9LP/vzrWmu6B+xoDVGhgaP+xJ2VlcXWrVvp\n7e0lLCyMTZs2UVJS4oHKjXXllVfyzDPPcP/99/PMM8+wcuVKo0tyK9dCg98Dlmite0e1k9ba5x7A\nF0NCQnROTo5+8MEHtS/ZvHmzBnRBQYEuKirSRUVFeuG/vaa/+uQ2o0tzm/v/sldP/f5r+gtXXmV0\nKW63a9cuPXfuXF1QUKBXrlypW1tbjS7JrX74wx/q6dOn65kzZ+qbbrpJ9/f3G13SuFx33XU6JSVF\nBwYG6vT0dL127Vrd0tKiL774Yj116lS9bNkyfeLECaPLPGfDvT/gKEPtzLtdj9/qEX7nyhQpPuCB\nl/fx0s5a9vzoUq9erhPgRPcA5695h1VzMnhole90IhDCi4y50ce7f+sIABZkx9Mz6OBAXafRpYzb\nuq1VDNid3H6Bb3T5FcIfSJD4gAXZQ20j2yu9e7qUfpuDbZUn+EpJBlOTIo0uRwgxShIkPiApOpTs\nhAi2VXr3Wl+v7qnjo2MnWFnsW73shPB1EiQ+YkF2HNsrWz0+DYanaK156sPjTE+O4vwpf79QjxDC\nvCRIfMT87Dg6+4fWdvZGWytaKa/v5LYLJvvsAC8hfJUEiY9YkBNPUIBi56feORPpUx9WEhcRLLe1\nhPBCEiQ+It0aRnZCBO8cajK6lDH79EQPG8sbuXFBFqFB3jU7rBBCgsSnnJcTz0fHWrxufZKnPzpO\noEVx08JJRpcihN9atWoVDzzwAEqpD5RSVUqpS0a7rwSJD7loehL9NqdXdQPu6rfxp9IarihMIzk6\ndOQdhBAesW/fPqxWK1rrxcA9wI2j3VeCxIcszIknONDCe4e8Z32SF0tr6B6wc9siGYAohFF6e3vp\n6OjgvvvuO/lSENA+2v0lSHxIWHAAC3Piee+wd7STOJyapz+qZN7kWAoyYowuRwi/VVZWxty5cwkI\n+KyNshDYP9r9DQ0SpVScUmqDUuqI68/YM2znUErtdj1emeg6vclF0xKpaO6hunV0k3YaaWN5I9Wt\nfXI1IoTB9u3bR3Fx8akvFQJ7R7u/0Vck9wObtNa5wCbX8+H0aa2LXY8rJ64873PR9KG1IN7zgt5b\nf/6khnRrGMvzk40uRQi/NkyQzMJbrkiAlcAzrq+fAf7BwFp8QnZCBFlx4aZvJ9lV1caGska+ftEU\nr5+xWAhv9/DDD3Pttdd+9lxrnaO17hvt/kb/BCdrretdXzcAZ/poGqqUKlVKbVVKnTFslFKrXduV\nNjeb+xeppyiluGh6Irtr2ukbNG834Cc+qCA6NHBoYSUhhFfzeJAopTYqpfYP8/jcsmJ6aGGUM00U\nNUlrXQLcAPyXUmrKcBtprZ/QWpdorUtOLvfpj76Qn0JHr42tFeacxLGypYc3DzRw83mTiAzxxUU6\nhfAvHv8p1lqfcVCLUqpRKZWqta5XSqUCw97Y11rXuv6sUEq9x9DioMc8Ua8vmJcdR1hQAG/ub2Dp\nDPOtJ/37zRUEBVi45fzJRpcihHADo29tvQLc4vr6FmD96RsopWKVUiGurxOARUDZhFXohYIDLSzL\nS+LtsgbsDqfR5XxOc9cAf/6khqvmZJAUJQMQhfAFRgfJGmC5UuoIcInrOUqpEqXUWtc2eUCpUmoP\n8C6wRmstQTKCFbNSaOu1sf24uUa5P/1RJTaHkzsvlC6/QvgKQ29Qa61PAMuGeb0UuMP19UeALN49\nRounJRIaZOGt/Q2cPyXB6HIA6B6w878ff8oX8lPISZQVEIXwFUZfkQgPCQ8OZMm0RN460IjTJItd\nPb+9is5+O3ctyTG6FCGEG0mQ+LDLC1JJs4ayq9r4NUpsDidPbqlkQXYcs7OGncBACOGlJEh82NIZ\nSRyo6+TlXXVGl8Kre+qo7+jn7iXD9twWQngxCRIfFhUaxKUzU3h1bx2DduN6b2mt+d37FUxPjvps\nChchhO+QIPFxX56dRnuvzdC5tzaUNxATFsQ3lk6R9diF8EESJD7uwtxE4iOCeXl3rSHndzo1v9pw\nlKaufi4vSDWkBiGEZ0mQ+LigAAtfKkpjY3kT7b2DE37+Nw80UF7fyT2X5MrkjEL4KPnJ9gPXlGSQ\nmxTJ+t0T2+jucGp+teEwUxIjuLJIJmcUwldJkPiBmWkxWJTiue1VDM2NOTFe31fPkaZu7r1kGgEW\naRsRwldJkPiJGxdkcbChi51VEzOmZNDu5KWdNVw0PVHaRoTwcRIkfuLK4jSW5CbwYmnNhJzv2Y+P\n8+6hZm45fzIWuRoRwqdJkPiJ8OBApqVE8edPaqhtH/XCZ+ektWeQRzYdYfG0RJZON9809kII95Ig\n8SO3nD8ZrTUv7qj26Hn+a+NhegcdPHB5nkfPI4QwBwkSP5IRG87N503i95sraO3xTFfgQw2d/GFb\nFTcuyGJacpRHziGEMBcJEj9z88JJ9Nkc/H5zhduP7XRqHny9nAumxnPvJdPcfnwhhDlJkPiZqUlR\n3LYom3fKm2jq7HfrsV8orWbzkRYum5VKXESwW48thDAvCRI/dPPCSVS0dPPwhsNuO2Ztex+v761n\n0dR4rp2X6bbjCiHMT4LED01OiOCbS6ey43gr+2s7xn08h1Nz3wu72VXVxk+/XCATMwrhZyRI/NRt\nF2TT2W/noTfKsTvGN8X8Ex8co6atlx9fOZNJ8RFuqlAI4S0kSPxUdGgQa1YVsLu6nUffPXrOx9lQ\n1sh/vnmIxbkJXD03w40VCiG8hQSJH1uWl8wXC1L58GgLHx5tGfP++2o6WLu5gsKMGH585Sy5pSWE\nn5Ig8XM/unImHX02fvv+MQ43do16vwO1Hdz81Dbq2vtY+9USQoMCPFilEMLMDA0SpdQ1SqkDSimn\nUqrkLNutUEodUkodVUrdP5E1+rrIkED+59Z5HG7s4j9eL2dfzciN7+8fbua6Jz6mZFIsf7hjIUnR\noRNQqRDCrIy+ItkPrAI+ONMGSqkA4DHgMiAfuF4plT8x5fmH9NhwXlh9HvUdfXz7+V08v70K+zBr\nvHf02vjF24e4/entLJqawL+tnEVWfLgBFQshzCTQyJNrrcuBke6tzweOaq0rXNs+D6wEyjxeoB+Z\nnBDButsX8LM3D7J2cwVvlzVSmBFDmjWUAZuTY83dHGro4khTN7ecn80/XTqdsGC5nSWEMDhIRikd\nOHWWwRpgwXAbKqVWA6sB8vPlomWskqJD+fk1Rbx7qJn1u2spPd5KXXsfn7b2MW9SLHMmWfnhl2aS\nlxptdKlCCBPxeJAopTYCKcN861+11uvdeS6t9RPAEyefuvPY/kIpxcUzkrh4RhK9g3ZaugYJDbKQ\nEBki64oIIYbl8SDRWl8yzkPUAqfOuZHhek14WHhwIFnx3nDRKoQwktGN7aOxA8hVSmUrpYKB64BX\nDK5JCCGEi9Hdf7+slKoBzgNeV0q95Xo9TSn1BoDW2g58C3gLKAde1FofMKpmIYQQn6e09tmmBJ99\nY0II4UFjbgz1hltbQgghTEyCRAghxLhIkAghhBgXCRIhhBDj4suDBGT0nBBCTAC5IhFCCDEuEiRC\nCCHGRYJECCHEuEiQCCGEGBcJEiGEEOMiQSKEEGJcJEiEEEKMiwSJEEKIcZEgEUIIMS4SJEIIIcbl\n/wMG54KvMAQDPAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f09015b85f8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eqn1 = eqn.subs('a', 4).subs('b', 1).subs('c', 2)\n",
    "soln = rsolve(eqn1, y(n), init={y(0) : 1, y(1) : 2})\n",
    "display(soln)\n",
    "\n",
    "plot(soln, (n, 0, 12), xlabel=\"$n$\", ylabel=\"$y_{n}$\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Interactive solution\n",
    "\n",
    "With an interactive plot, we can see how the form of the solution changes with changing constants in the difference equation.\n",
    "\n",
    "We first define the difference equation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAATMAAAAUBAMAAAATqMdXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nhA/XnkG39AAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$a y{\\left (n \\right )} - b y{\\left (n - 1 \\right )} + c y{\\left (n - 2 \\right )} = 0$$"
      ],
      "text/plain": [
       "a⋅y(n) - b⋅y(n - 1) + c⋅y(n - 2) = 0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = a*y(n) - b*y(n - 1) + c*y(n - 2)\n",
    "eqn = Eq(f, 0)\n",
    "display(eqn)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solving and plotting:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "3cd83f92cd3246e3bb54d54abec9acbe",
       "version_major": 2,
       "version_minor": 0
      },
      "text/html": [
       "<p>Failed to display Jupyter Widget of type <code>interactive</code>.</p>\n",
       "<p>\n",
       "  If you're reading this message in Jupyter Notebook or JupyterLab, it may mean\n",
       "  that the widgets JavaScript is still loading. If this message persists, it\n",
       "  likely means that the widgets JavaScript library is either not installed or\n",
       "  not enabled. See the <a href=\"https://ipywidgets.readthedocs.io/en/stable/user_install.html\">Jupyter\n",
       "  Widgets Documentation</a> for setup instructions.\n",
       "</p>\n",
       "<p>\n",
       "  If you're reading this message in another notebook frontend (for example, a static\n",
       "  rendering on GitHub or <a href=\"https://nbviewer.jupyter.org/\">NBViewer</a>),\n",
       "  it may mean that your frontend doesn't currently support widgets.\n",
       "</p>\n"
      ],
      "text/plain": [
       "interactive(children=(IntSlider(value=6, description='a', max=11, min=-10, step=2), IntSlider(value=1, description='b', max=20, min=-2), IntSlider(value=8, description='c', max=10, min=1), Output()), _dom_classes=('widget-interact',))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_solution(a=6, b=1, c=8):\n",
    "    # Substitute in parameters\n",
    "    eqn1 = eqn.subs('a', a).subs('b', b).subs('c', c)\n",
    "    \n",
    "    # Check roots\n",
    "    roots = ((-b - sqrt(b**2 - 4*a*c))/(2*a), (-b + sqrt(b**2 - 4*a*c))/(2*a))\n",
    "    print(\"Roots of characteristic eqn:\")\n",
    "    display(roots)\n",
    "\n",
    "    # Solve\n",
    "    soln = rsolve(eqn1, y(n), init={y(0) : 1, y(1) : 2})\n",
    "    print(\"Solution:\")\n",
    "    display(soln)\n",
    "    \n",
    "    # Plot\n",
    "    plot(soln, (n, 0, 20), xlabel=\"$n$\", ylabel=\"$y_n$\", axis_center=\"auto\", adaptive=False, nb_of_points=21)\n",
    "\n",
    "    \n",
    "interact(plot_solution, a=(-10, 11, 2), b=(-2, 20, 1), c=(1, 10, 1), continuous_update=False);"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}