{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sympy : Symbolic Mathematics in Python\n",
    "\n",
    "http://www.scipy-lectures.org/packages/sympy.html\n",
    "\n",
    "![](./img/sympylogo.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "x = Symbol('x')\n",
    "y = Symbol('y')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Integration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x**3, (x, -1, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "log(x)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate( 1/x )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "log(2*x + 1)/2"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate( 1/(1 +2*x) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-log(2*x - 1)/2"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate( 1/(1 - 2*x) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Equation solving"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{x: -3, y: 1}"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve([x + 5*y - 2, -3*x + 6*y - 15], [x, y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Differential Equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "f, g = symbols('f g', cls=Function)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f(x) + Derivative(f(x), x, x)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x).diff(x, x) + f(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Eq(f(x), C1*sin(x) + C2*cos(x))"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsolve(f(x).diff(x, x) + f(x), f(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Eq(f(x), -asin(sqrt(C1/(sin(x)**2 - 1) + 1)) + pi),\n",
       " Eq(f(x), asin(sqrt(C1/(sin(x)**2 - 1) + 1)) + pi),\n",
       " Eq(f(x), -asin(sqrt(C1/(sin(x)**2 - 1) + 1))),\n",
       " Eq(f(x), asin(sqrt(C1/(sin(x)**2 - 1) + 1)))]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsolve(sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x), f(x), hint='separable')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1/x**2"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(1/x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1/(2*y**2)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(1/(2*y), y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1/(2*y**2) - 1/x**2"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(1/x, x) - diff(1/(2*y), y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from __future__ import division\n",
    "from sympy import *\n",
    "x, y, z, t = symbols('x y z t')\n",
    "k, m, n = symbols('k m n', integer=True)\n",
    "f, g, h = symbols('f g h', cls=Function)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Eq(Derivative(x(t), t), 12*t*x(t) + 8*y(t)),\n",
       " Eq(Derivative(y(t), t), 7*t*y(t) + 21*x(t)))"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq = (Eq(Derivative(x(t),t), 12*t*x(t) + 8*y(t)), Eq(Derivative(y(t),t), 21*x(t) + 7*t*y(t)))\n",
    "eq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Eq(x(t), C1*x0 + C2*x0*Integral(8*exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0**2, t)),\n",
       " Eq(y(t), C1*y0 + C2(y0*Integral(8*exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0**2, t) + exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0))]"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsolve(eq)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Eq(Derivative(x(t), t), 12*t*x(t) + 8*y(t))"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq1 = (Eq(Derivative(x(t),t), 12*t*x(t) + 8*y(t)))\n",
    "eq1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Eq(-4*Integral(2*y(t)*exp(-6*t**2), t) - 4*Integral(3*t*x(t)*exp(-6*t**2), t), C1)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsolve(eq1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Eq(Derivative(1/x, x) - Derivative(1/(2*y), y), 0)"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq1 = (Eq(Derivative(1/x,x)- Derivative((2*y)**(-1), y)))\n",
    "eq1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "The function cannot be automatically detected for Derivative(1/x, x) - Derivative(1/(2*y), y).",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-57-b4355e87dcda>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mdsolve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0meq1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/Users/chengjun/anaconda/lib/python2.7/site-packages/sympy/solvers/ode.py\u001b[0m in \u001b[0;36mdsolve\u001b[0;34m(eq, func, hint, simplify, ics, xi, eta, x0, n, **kwargs)\u001b[0m\n\u001b[1;32m    623\u001b[0m         hints = _desolve(eq, func=func,\n\u001b[1;32m    624\u001b[0m             \u001b[0mhint\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msimplify\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mTrue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mxi\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mxi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0meta\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0meta\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m'ode'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mics\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mics\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 625\u001b[0;31m             x0=x0, n=n, **kwargs)\n\u001b[0m\u001b[1;32m    626\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    627\u001b[0m         \u001b[0meq\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mhints\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpop\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'eq'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0meq\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/chengjun/anaconda/lib/python2.7/site-packages/sympy/solvers/deutils.py\u001b[0m in \u001b[0;36m_desolve\u001b[0;34m(eq, func, hint, ics, simplify, **kwargs)\u001b[0m\n\u001b[1;32m    173\u001b[0m     \u001b[0;31m# preprocess the equation and find func if not given\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    174\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mprep\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mfunc\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 175\u001b[0;31m         \u001b[0meq\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfunc\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_preprocess\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0meq\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    176\u001b[0m         \u001b[0mprep\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mFalse\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    177\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/chengjun/anaconda/lib/python2.7/site-packages/sympy/solvers/deutils.py\u001b[0m in \u001b[0;36m_preprocess\u001b[0;34m(expr, func, hint)\u001b[0m\n\u001b[1;32m     75\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfuncs\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     76\u001b[0m             raise ValueError('The function cannot be '\n\u001b[0;32m---> 77\u001b[0;31m                 'automatically detected for %s.' % expr)\n\u001b[0m\u001b[1;32m     78\u001b[0m         \u001b[0mfunc\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfuncs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpop\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     79\u001b[0m     \u001b[0mfvars\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mset\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mValueError\u001b[0m: The function cannot be automatically detected for Derivative(1/x, x) - Derivative(1/(2*y), y)."
     ]
    }
   ],
   "source": [
    "dsolve(eq1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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