12-computacao-simbolica.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Computação simbólica\n",
    "====================\n",
    "\n",
    "SymPy\n",
    "-----\n",
    "\n",
    "Nesta seção, apresentamos algumas funcionalidades básicas da biblioteca SymPy (SYMbolic Python). Em contraste com a computação numérica (envolvendo números), no cálculo simbólico estamos processando e transformando variáveis genéricas.\n",
    "\n",
    "A página inicial do SymPy (<http://sympy.org/>) fornece a documentação completa (e atualizada) para esta biblioteca.\n",
    "\n",
    "A computação simbólica é muito lento em comparação com as operações em ponto flutuante e, assim, geralmente não muito adequada para simulação direta. No entanto, é uma ferramenta poderosa no suporte à preparação do código e trabalho simbólico. Ocasionalmente, usamos operações simbólicas em simulações para elaborar o código numérico mais eficiente, antes que seja executado.\n",
    "\n",
    "### Saída\n",
    "\n",
    "Antes de começarmos a usar o SymPy, chamaremos `init_printing`. Isto diz ao SymPy para mostrar as expressões em um formato mais conveniente."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy\n",
    "sympy.init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Símbolos\n",
    "\n",
    "Antes de começarmos a executar qualquer operação simbólica, precisamos criar variáveis simbólicas usando a função `Symbol` do SymPy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.symbol.Symbol"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import Symbol\n",
    "x = Symbol('x')\n",
    "type(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAJBAMAAAAWSsseAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEHarIkSJZt3NVLsymT3i\nYlMAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABASURBVAgdY2AQUnZVU2BgTGBv4pjAwCbA9pDLgYGR\ngXMDAwjwKYCpcweAFJeAHgOTAEPcgn0M7gwMwkpC1wsYAB0ECeuXDPmiAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$x$$"
      ],
      "text/plain": [
       "x"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = Symbol('y')\n",
    "2 * x - x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAASBAMAAAAHyPWiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABIklEQVQoFWNgVDJ2YCAChDGw/8CqjHcqUBhM\ngKVTGRhmIqvjPQDhhal9ZWAAExD+LAaGfAcIE0zC1DHwANVBCLD4/gDi1AEVrw9gPZPYUwPWxoBp\nXowOAyPIrRzfGcIZXh1YjUtdwU0Gng9ASa4EhlKG+oCLONQxXtBmYDYASioxMAQwbAarYt+9e9fs\n3bsTQByYP1gZ5jDwHwDyFUCif0AECGC6j+Mjg78DA8MlBkYBBpYvYEVY1TFtYDjJwMCiwMAjcBXo\n0KMQlZjmMScwmDEwSJUXZXH8Zf7A+ACXOiYDjn8MDPv////FmnP00DuIMrj7mDJ/5zGACdb0Qx+h\nkkgU3F4kMaALMQCrALpQNkN8A7oYNv46BhtswhhiPYccGBgAUNlc2Xs0Jf4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$2 x + 11 y$$"
      ],
      "text/plain": [
       "2⋅x + 11⋅y"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x + y + x + 10*y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAQBAMAAABelcpIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2UlEQVQYGWNgwAEYr5UdwC7F9ZCBgXMBYwuK\nLJcChCt8+SMDQx0DQyZWWQYmoOxmBob1AsjSML1g2c8MDPELGBiElF3VFMCKkGUZ/wJlFRgYE9ib\nOCZgyLJ8Y2A4n8DAJsD2kcsBUxaoFyjLyMC5ASTHNnPmjJczZxqA2EBXQU1mYOBXAIkAAbK9DEBX\nrV8AFDx/AEiAAIrsbgaGegEGLgF9BiaIx1BkgaFxE+Sp/QzuWPRyLGB8w8AgrCR0vQBVltfupzkD\nY5LyAYgwlISbjCIK4zA6wFgwGgBa/zSme/c4VQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x + 10$$"
      ],
      "text/plain": [
       "x + 10"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y + x - y + 10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos abreviar a criação de múltiplas variáveis simbólicas usando a função `symbols`. Por exemplo, para criar as variáveis simbólicas `x`, `y` e `z`, podemos usar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEIAAAASBAMAAAAd9BUsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABQklEQVQoFW2Pv0vDQBTHv5cfmJZoAg6CUwg4\n6RAQxUEhf0GNCE5Cs4igg+JSVJDg4OQqqDi4SZfaSYQMLd0UB3EThHZ1M4MoQq3vvGu0Id/h7vs+\n97337sDcaR85GqssSLqCoc+cAHPQ7gi+Bpz9T7yIwuzCuhT2HNjyhf1dZcI4RbkucCPITdBhfwrZ\n62B1CuxE3JA9AP0KMKpRibDxge1nmG+DCbY3ASw/uB7hYsgeJ6FyS0p7YDdCgFHOXOi4gNUkux/H\n1Ti+4RQo0Fxthozp0KAEiz55kuzBbKhd4IijJzBbqeOOe5JMWO88MRwS0RyYthpiljyXTBQ8FBO6\njRDjlZ0NKJ7xLQL9xEgH7aZWO7iP0Oj1vqCvt5LBBA7dOSh4vZUYoJcI/f02PeNmE+VIgiW5Z7Ya\n5jMkWx63/CxK6x97OUvyEV1R8gAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2 y + 3 z$$"
      ],
      "text/plain": [
       "2⋅y + 3⋅z"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy\n",
    "x, y, z = sympy.symbols('x,y,z')\n",
    "x + 2*y + 3*z - x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Uma vez que terminarmos a manipulação dos termos, às vezes desejamos inserir números para as variáveis. Isso pode ser feito usando o método `subs`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADsAAAASBAMAAAAXqLt+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABB0lEQVQoFWNgwAJ8b+/BIgoTYjRgyA+AcZDo\nRRA2swAD3wMkYRgTKs3XwMD8ByaGREOlOT8iSQspu6opQNRApYEcjo+MugoMfgIMjAnsTRwT0KXj\nE9i4HjDoMzCwCbB95HJAl25m8OKdwLCbgYGRgXMDWDJz5sy+mTMng9m8FxgE+BQYPoE4/ApgISAB\nt1sXyIl34AE7//wBdGmuCwwuDDcZmB4wMHAJ6DMwCUAUwHSfZWBQYdjMwG0ANGPBfgZ3qHaoNEtb\n2uUGhtsM8QUMDMJKQteBFBhApTn////fwCCcdN8BKg6lYIZDuTdRZRlEkPjMExi/InHRmRwF3Aro\nYkh8VqU7QB4ArNM4RQcXfGIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x + 2 y$$"
      ],
      "text/plain": [
       "x + 2⋅y"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import symbols\n",
    "x, y = symbols('x,y')\n",
    "x + 2*y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADsAAAASBAMAAAAXqLt+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABB0lEQVQoFWNgwAJ8b+/BIgoTYjRgyA+AcZDo\nRRA2swAD3wMkYRgTKs3XwMD8ByaGREOlOT8iSQspu6opQNRApYEcjo+MugoMfgIMjAnsTRwT0KXj\nE9i4HjDoMzCwCbB95HJAl25m8OKdwLCbgYGRgXMDWDJz5sy+mTMng9m8FxgE+BQYPoE4/ApgISAB\nt1sXyIl34AE7//wBdGmuCwwuDDcZmB4wMHAJ6DMwCUAUwHSfZWBQYdjMwG0ANGPBfgZ3qHaoNEtb\n2uUGhtsM8QUMDMJKQteBFBhApTn////fwCCcdN8BKg6lYIZDuTdRZRlEkPjMExi/InHRmRwF3Aro\nYkh8VqU7QB4ArNM4RQcXfGIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x + 2 y$$"
      ],
      "text/plain": [
       "x + 2⋅y"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x + 2*y.subs(x, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAPBAMAAAD0aukfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWaJMkS7\nIt2ZnNffAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAcElEQVQIHWNgAAPOabOANKMyAwNzNUM4A4NJ\nyGcGBp4AhjNAUTYgc/8FsDIQ8xCYBRb9vjJ1A0QB8/cGBkUo878BQ5ABRNs/BoZ+oAqQtudAZgOE\n6YQQXQ9UC9HGwDKBGWgCq9N3DwaGqSELgKJwAADSWSGR+xZb5QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$16$$"
      ],
      "text/plain": [
       "16"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x + 2*y).subs(x, 10).subs(y, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAPBAMAAAD0aukfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWaJMkS7\nIt2ZnNffAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAcElEQVQIHWNgAAPOabOANKMyAwNzNUM4A4NJ\nyGcGBp4AhjNAUTYgc/8FsDIQ8xCYBRb9vjJ1A0QB8/cGBkUo878BQ5ABRNs/BoZ+oAqQtudAZgOE\n6YQQXQ9UC9HGwDKBGWgCq9N3DwaGqSELgKJwAADSWSGR+xZb5QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$16$$"
      ],
      "text/plain": [
       "16"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x + 2*y).subs({x:10, y:3})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Também podemos substituir uma variável simbólica por outra, como neste exemplo, em que `y` é substituído por `x` antes de substituirmos `x` pelo número `2`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEIAAAAYBAMAAAC8bzZKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIom7VJlmdt1E7xDN\nqzIhoty3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABPklEQVQoFWNgwAtUTTfglWdgEuA9gF8FhwDf\nRwIqJjD+xK+CgYHlNyEVHAUMDEJKJniUqTIwMAYw+CfgVMIWwMDA/oWBvwFZhTsyZxLDPAbOlQz7\nDyALIqvgO+JiDZJDtQVZBdf//x+ACribGBi210oWAV0NAlAV28MZGFdDRBg1ohm4FdjaOQ5A+FAV\nCl4M7CD9YKA1gWkD028WAQgPooLRIYaB2QCqgIFrNTcDD4Snc+ZM15kzZ4FWMyxn4C8Aq2AUYGD+\nwgDjAYWgtnB+ZHj/AKyC/zdYBZSHUMF6gKEKrICBy4CB4yPLhnoG9g0QAagZzBcYbCECbAkM/gXz\nHewZxCB8mC2sBpzfoCJTQy0Z9pZvD0tAVcF9qRRX4oHawsAAdAl2sBEifIdh/wTsCmCibQzmMCYO\nenrpA4gMADQxRJY5586uAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$3 x + y^{2}$$"
      ],
      "text/plain": [
       "       2\n",
       "3⋅x + y "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "termo = 3*x + y**2\n",
    "termo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEMAAAAYBAMAAABTrV10AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHaZIu+JVM27RDKr\nZt2dj8xZAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABHklEQVQoFWNgAAM2S0MIAzeZwnACtyRE5gnD\nfgcCal4x+C8goISBoV6AoJJHBFWwGaApcUubjiZyGY3PYsBQfwFFjCuBawNEoAhCMX1n4H+AomRH\neSXUuVAlXFEM8g0oSuL//4fyoUqAPLhFjMoKDK5QE0Cq4EoYnzFA5djYAxj0oSYgK2HJM2SAyrkz\nNzDMxKKEgSF1AVROgFWB4RNMSUZHx9OOjh4IlzUQJrfegfs3TAmQhrqFxYGB8zsDVM6CgSkAUwn/\nR5ASqNwcBs4DmEpYJzDwfWGAylkyrL+AqYT3AkO9AgNUTibJ3gFTCcNi41kMDHA5CyQVMOfChEBy\nPA2Mv2B8EC2CxIHK8V3gVEASRWFC5biUrFGEkTkQOQCrJ0MHUC4xWQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$y^{2} + 3 y$$"
      ],
      "text/plain": [
       " 2      \n",
       "y  + 3⋅y"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "termo.subs(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAPBAMAAAD0aukfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWaJMt0i\nu0SCRuA9AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAcUlEQVQIHWNgAAHmyM4FQIpRmYGBXYB5DwOD\nSchnBoYmBobpQFE2IPMKA4O8AYT5hYHhvQCYyfwVyEwAM3l+MjCsnwBhAkWhTCQFDEBt8hBtDHcZ\nGPqhhgGtCIdawS3ArMXAwOr03YOBeVrKAqAoHAAARgMh+ZrNuUcAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$10$$"
      ],
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "termo.subs(x, y).subs(y, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A partir deste ponto, alguns fragmentos de código e exemplos que apresentamos assumirão que os símbolos necessários já foram definidos. Se você tentar um exemplo e o SymPy der uma mensagem como `NameError: name 'x' is not defined`, é porque você precisa definir o símbolo usando um dos métodos acima."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### isympy\n",
    "\n",
    "O executável `isympy` é um _wrapper_ em torno do ipython que cria as variáveis simbólicas (reais) `x`, `y` e `z`, as variáveis inteiras simbólicas `k`, `m` e `n`, e as variáveis de função simbólica `f`, `g` e `h`, e importa todos os objetos do SymPy.\n",
    "\n",
    "Isto é conveniente para descobrir novos recursos ou fazer experimentos de forma interativa\n",
    "\n",
    "   $> isympy\n",
    "    Python 2.6.5 console for SymPy 0.6.7\n",
    "\n",
    "    These commands were executed:\n",
    "    >>> from __future__ import division\n",
    "    >>> from sympy import *\n",
    "    >>> x, y, z = symbols('xyz')\n",
    "    >>> k, m, n = symbols('kmn', integer=True)\n",
    "    >>> f, g, h = map(Function, 'fgh')\n",
    "\n",
    "    Documentation can be found at http://sympy.org/\n",
    "\n",
    "    In [1]: "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Tipos numéricos\n",
    "\n",
    "O SymPy tem os tipos numéricos `Rational` e `RealNumber`. A classe `Rational` representa um número racional como um par de dois inteiros: o numerador e o denominador, então `Rational(1,2)` representa `1/2`, `Rational(5,2)` representa `5/2`, e assim por diante."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import Rational"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAArBAMAAABlSd54AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWaJMt0i\nu0SCRuA9AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAg0lEQVQoFWNgQAdCBjARZld/OJuBIX9g2CH6\nJQ0wB9GS/g8HH6hgjemMWWBTQIFZyGB9gIEBHJgcCgycBSAJYGDyOjAwfoay+R0Y+D5C2fYBDHzf\noOzzExi4fsLYAQg2shqgXkaYXt4LDBwwM4F2scLsYtjIYLQAaA44MC3TvEBGIgAAytYwIjgl9mIA\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{1}{10}$$"
      ],
      "text/plain": [
       "1/10"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = Rational(1, 10)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAAqBAMAAACuFQ3dAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMpndu3bvImbNiRBU\nq0Qb3U6NAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCElEQVQYGbXRv0rDUBTH8W+utQ2xf0KfQB0V\npIOrUFz7BDpl6KZgcXLTRecIFcSpgy4BwUVBF/MCQjsUilDo4i4iWiwaf/cW9AHES+7Nh3MP55AT\ngCqcXFVTCVZhJ/twnG3CWb/lvNeGrhN0fu235MHhgr25Rm6wP5I71pCrQTGc2rzDHXIpJviCh+Hw\nrZ+PMRObSRMDM7HzK4V5jlzj82zA7uWyC//vkf2s5z82SjZDb20pSVTmdmRiX4U/5XVydaOP6kJZ\nsyDQE2pGkaz1pF05vlixrmvfn5JPlXKAPMFs6SVSqeFrfBvWmpc/dv8MSpGNz9lOeMqP8F6seeQm\npTh1sNiDwrai37HIT2/0Bu2KAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{45}{67}$$"
      ],
      "text/plain": [
       "45\n",
       "──\n",
       "67"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = Rational(45, 67)\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACAAAAArBAMAAADrk5cEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJqyLvZs27djJE\nVN3lLQH2AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABAUlEQVQoFWNgAANGZZMACAtKihWwpqIIuDEw\nOKEI/GBgsEcWYPzJwLB+A7JIBlDFAWQBoPr+AmQBjgW89heQBRiqVtujmAGU9BdAUcHA0ILCLwvg\n/ooisEJAAsUSBmYlHRQFg4DzHw3Q0knBxnNBxm8C23FNgIFVgeE+KIBeAjGjEzB0uf4y8D9gYODK\nBKtYL8DAk8Mg38DAEN4JEwDSIC0GSAKMjxkY2AMQAqx2igwMUgwIAQYGswMMBigCbIm8GxACrAEM\nnH+FGRAC/B+BAldcXL55gewFuoNtAgMfKG4zQXyQAHsBw/0FQCYwqTEwqOS3FDAcUpoDZFr+dweJ\noAIARCNKJ4JmHCEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{9}{134}$$"
      ],
      "text/plain": [
       "9/134"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a * b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADEAAAAqBAMAAADoo653AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSwupteZCE6Z2+2DMgpk5EfcNOYuH+cuPnqMAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- \\frac{383}{670}$$"
      ],
      "text/plain": [
       "-383 \n",
       "─────\n",
       " 670 "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a - b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACAAAAAqBAMAAAAgz0ShAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{517}{670}$$"
      ],
      "text/plain": [
       "517\n",
       "───\n",
       "670"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a + b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note que a classe `Rational` funciona com expressões racionais *exatas*. Isto contrasta com o tipo de dado `float`, padrão do Python, que usa a representação em ponto flutuante para \n",
    "*aproximar* números racionais.\n",
    "\n",
    "Podemos converter o tipo `sympy.Rational` em uma variável de ponto flutuante no Python usando `float` ou o método `evalf` do objeto `Rational`. O método `evalf` pode levar um argumento que especifica quantos dígitos devem ser calculados para a aproximação de ponto flutuante (nem todos podem ser usados pelo tipo de ponto flutuante do Python, evidentemente)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAqElEQVQYGZ2PPQ4BURSFvyEm3vgZNNoXQq1Q\naDRWYAcsQTehMbEBGolyKtNaAjYyswQSU06e+94KxC3Od3PuOcWF3nAK1NbsY/CXqBOoBP8DjcJB\nzkEhAseNw9VpRTs8nbY0XVkW8AAvjcaJ1I0xgn9H2sa8fmr3o7nkPE2WQ7MkTKB+YXXDjjXlw9Sq\ntxtZwDZ2UGfxOlRLCAsHNSF4Qzsnu0vkMJjBFyxOKQHdWH1vAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{2}{3}$$"
      ],
      "text/plain": [
       "2/3"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = Rational(2, 3)\n",
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALQAAAAOBAMAAAB0sisSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n3C+0iL9XCwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$0.6666666666666666$$"
      ],
      "text/plain": [
       "0.6666666666666666"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "float(c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKsAAAAOBAMAAACiF/F3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$0.666666666666667$$"
      ],
      "text/plain": [
       "0.666666666666667"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c.evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAhcAAAAOBAMAAABusSMBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAACM0lEQVRIDbWWwUpbQRSG/5vcEGsMSRVKySah\nQhdSITTtNuQNDPQBzELpquAjuLILs/ANGlfZhi51E5BuXOUJFFroUohK2oKanjkRznhkcCaMgfsH\n8t3//8lh7twBlj98BH8av9qAkmyTIIubaCwef8K1YoxRFhSZ6642GjSFz3h9YIbxapTpaEm+4C1Y\n3ERj8fgT7hZjjLKwyHQ6nd4AaQ/ZDs0C31EYaCnUcAwWN9FYPP6Eu8UYoywsMkPrYQgUhsiPaRbF\nf48F52bNsChsEYXnIRxuGSOUBUZmaTOgTaI0RPEP/eVC77HgiH6bicIW4XsEz0PYbRklTRPvMm18\nIpL+5hld1RqKd/Rd2m6ua8Hk08/RTBS2CN8jeB7CbssoaZp4l2njE5E0gAFd3+rIXZmhbGFxR0ky\naaHPorEQjechXCvGGGWBkfSUtMwwavfDGCNzWH0oybSMtZdGyk6isXj8CYeLMUZZYCQtBpqFPCa7\nSG9LDwV/gY0Ri5sobHm8CYdbxghlgZHAbzMM2kDzZgNd6CC9VoI9GkaLxU0UtjzehMMtY4SywEhg\n3wyDDhepebUu9WhlKMGpWRksbqKw5fEmHG4ZI5QFRiKhIxfMoYtOnkCOtoueEmzSdjETN+F7BFse\nb8JuyyhpmjxXJHKXZhj4iko7mQAnqOxoWaonfbC4icbi8SfcLcYYZYGR+dkwVi5+AF16t7yhbyV4\n/47OZSxuorDl8SYcbhkjlAVGvujTuvgPtfGC8KpbxVAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$0.66666666666666666666666666666666666666666666666667$$"
      ],
      "text/plain": [
       "0.66666666666666666666666666666666666666666666666667"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c.evalf(50)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Diferenciação e Integração\n",
    "\n",
    "O SymPy é capaz de executar a diferenciação e integração de muitas funções:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAUBAMAAADFBIheAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJVCLvMs1Edmar\n3bti/yyrAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABN0lEQVQoFWNgwADRUJGtGDJAAdYHUFGeACzS\nHDBBxgYsspvgYlZwFoJxFc4Ug7PgDNYLcCbHBAYGIWMVBkarxRMYqnavAEpwKTAwMK46OgtoLLMC\nA2sKQ1lAbQDDK8YHDLxAWb4NDAzCDHsX3GBg4G5g4Ghg8C+4zMBQX5nIwAWU5XVgYDBksBdwZWBg\nucAgDzSJ/QsDA7/B/Vcgv7ABLRNg6AOyGNg/MMxfAFT0Eyj7QLj/E1SWgeEXVBauV1WAYT9QH8hk\nBvbvIFnOCyB7GRySGBjmT1nAwAN0EchVjlwfGBaCXcWazcAcUFbA0MySyMAXAPYG5xe+D6xARUwG\nDAwyWqYMrMo2EzitV1tCjGM8tXDRbiCTowBIoAFESMaiyYC4+GMBZDkYYI1BeOzDlUFVQyiklAMA\n9mhEQ5Gu6cwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "cos(x)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import Symbol, exp, sin, sqrt, diff\n",
    "x = Symbol('x')\n",
    "y = Symbol('y')\n",
    "diff(sin(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVElEQVQIHWNgEDIxZWBgSGeQmMDAsoCBOYGB\n+wAD+0cG/gMMvN8Z5BUYeP8xzDdgYP3MMF8BREJEgLLs3xm4NzCwfATpYkpgYGhnkApgYBB+d5QB\nAPogE3QldevOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(sin(x), y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH0AAAAWBAMAAAAfnY6PAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAhWMl0cKRo4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$9 x^{8} + 20 x + 3$$"
      ],
      "text/plain": [
       "   8           \n",
       "9⋅x  + 20⋅x + 3"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(10 + 3*x + 4*y + 10*x**2 + x**9, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAPBAMAAAAv0UM9AAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAMpndu3bvImbNiRBUq9OB\nhjcAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABESURBVAgdY2BgYBACYgYGExDBmgIiK6aAyAUgkqMA\nRG5lAJELQCSPAIjcxQAiz969++wqUIIBrIvhCYi55N0NEMXAAABbkhBrtxdTYQAAAABJRU5ErkJg\ngg==\n",
      "text/latex": [
       "$$4$$"
      ],
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(10 + 3*x + 4*y + 10*x**2 + x**9, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAOBAMAAADd6iHDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIom7VJlmdt1E7xDN\nMqsI8sYEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAj0lEQVQIHWNgEFIyYWAQDbZlYGAMYPBP4H7A\noD+Bgf0LA38D3wYGrgUMnCsZ9h/gamDg+8sABP4JPL8hTO4mEJ/jN1CfRjSIOV8BRGpNABJ9IBYD\n12oGBtYAoLwAA/MXBoZSoBD/bxCTJYBBkIHLgIHjI8NVBoZCBrYEBv8CxialuAYGhqmhlgw8////\nbwAA/SMgBrNAMgkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$32$$"
      ],
      "text/plain": [
       "32"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(10 + 3*x + 4*y + 10*x**2 + x**9, x).subs(x,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHUAAAAPBAMAAAAok50oAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nwOj6ZhPluVM8xcoEeGPTfMx/rL8eWcW9C0KifwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$263.66015625$$"
      ],
      "text/plain": [
       "263.660156250000"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(10 + 3*x + 4*y + 10*x**2 + x**9, x).subs(x,1.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAQBAMAAAAG6llRAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEIl2mSJE3e9UMqtmzXCQ\nkgMAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABlSURBVAgdY2AAA2Uv6wUQFrPAXvYECJOL4Q6EASKf\nwplcD7ig7DimAiYoU9pMWRHIZFQ2CYCKpAewKkCYrJ0MIgsgTO4e44MQFgPfBiiDgYFPAc7kADIN\nIDzmCwwiMMNMbA5ABAE+RA6ieYoxNgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$e^{x}$$"
      ],
      "text/plain": [
       " x\n",
       "ℯ "
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(exp(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEUAAAAaBAMAAAATe4w4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMnarIkSJZlS7\nme8N5bApAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+ElEQVQoFWNgwAtYZh7HKw+S5GFoJqiGiUFe\ngKAihs2ElTCqE1YjRtgqxuuSeM1hj4xKYP7/Ga8abo4DE/AqAEmyBhBUwsnTsICQIralUx0IqcEr\nv+cMEAgw/IeCD3gVM3jOKcCvACh7gMOAoBoFxo/IahhD01unMjBIhl5HFuVGMUeSYXGAEQOHOUMl\nshohFPdMZJgvsJ2hbgPjFSQ1jIlgjpAxCKgwCDDoAPl5S6MEkNTIFk5A4gGZX4H4G6qQ/X8U9zCw\nfAHK/0RVg8bbyPqBIZDhJTC20SQQXI7fPB/YFzAEMTCi+B2hAMhiTA2MXQU05CqQQAcAqCA5Th40\nCJEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- x e^{- \\frac{x^{2}}{2}}$$"
      ],
      "text/plain": [
       "      2 \n",
       "    -x  \n",
       "    ────\n",
       "     2  \n",
       "-x⋅ℯ    "
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(exp(-x ** 2 / 2), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A função SymPy `diff()` usa no mínimo dois argumentos: a função a ser diferenciada e a variável em relação à qual a diferenciação é realizada. Derivadas de ordem superior podem ser calculadas pela especificação de variáveis adicionais, ou pela adição de um argumento inteiro opcional:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACcAAAAVBAMAAAA3P+qgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWYiuzKJ\nRN0MreaOAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0ElEQVQYGWNgQAM8qc5oIkBuBEMUpmAMQz+m\nIAM2lQzM2lhUpidABTlDV22Ay4MsYlRmYDBh4PkLF5wvwGAS8pmBAeg0JYgg0MT5DQwMbEBBPQYG\nf6j+lwxJBhDBegO44J4cT6AekEogeG/AnOZ57RiYDRPk/cNgw9CX8BxFkM+B4TDDeQNJFMFQBgYD\nhiqoEFQ7WwCI/w1VUJSBcwID928UQe4ABrYJwmwfGJKhwiAnzTtz1I/3I9cHzgaIIKvTdw+G+v//\n/zG7Jad2wPVjMgDBETUVhn8LIAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$12 x^{3}$$"
      ],
      "text/plain": [
       "    3\n",
       "12⋅x "
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(3*x**4, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB4AAAAOBAMAAADKyLEKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMt2rmYlmIkR2uxDN\nVO+L8+I6AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAt0lEQVQIHWMQMgkNDWB3WSXAAAH5////N9jM\nwPoDyndnYGBmsGZgeAflBzAwBDI8YmCwh2lg4JjAcH4DEp8LrFJ/A/dM885IIDsMxOf4yrCVIWmC\nDgMD9wcQn8mAIZohfsN1BgbOAyC+CwPDBoZTIBbTAiDB6QAkGH6CiP0GQOIKA3sBA+NHEF/egYGB\n0YGBs+A25weG2QwM94H8otAQI46/XB/YExgY6i4wMJz///8Tt+XsKWkMAEgNLU6wzhbHAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$72 x$$"
      ],
      "text/plain": [
       "72⋅x"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(3*x**4, x, x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB4AAAAOBAMAAADKyLEKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMt2rmYlmIkR2uxDN\nVO+L8+I6AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAt0lEQVQIHWMQMgkNDWB3WSXAAAH5////N9jM\nwPoDyndnYGBmsGZgeAflBzAwBDI8YmCwh2lg4JjAcH4DEp8LrFJ/A/dM885IIDsMxOf4yrCVIWmC\nDgMD9wcQn8mAIZohfsN1BgbOAyC+CwPDBoZTIBbTAiDB6QAkGH6CiP0GQOIKA3sBA+NHEF/egYGB\n0YGBs+A25weG2QwM94H8otAQI46/XB/YExgY6i4wMJz///8Tt+XsKWkMAEgNLU6wzhbHAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$72 x$$"
      ],
      "text/plain": [
       "72⋅x"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(3*x**4, x, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFEAAAAYBAMAAABw9gyhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWYyiSK7\nRN2jbGmSAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABoElEQVQoFY2SsS8DURzHv3XttXetw2BvGAwS\neUNj1RBhokJTROIS2oGh5y9gNBgkbAZNqMHiSCSWIsEm1GAgkW6MJBKCcN6997umhl686dP3+fS9\n310OqL+KF4zk+sZy/QzQ7aBFvuIk/UrD1r7I36z5hTDK0W8KHnxDLgOvVNzmRv1bwyRfwgpReEYb\nkRhNX67SJlCsEjZtfkU70OA4BUkJ6N5wiKS8sg89ZSTG+DDh7XlI6gPaPH+PO8JOnDBA5WVQ7LjU\nBQzT9dpCyxmVW2KQv+Uiq5Z8pBcq9WKBkyiXs/wgl/h6ZuieLc2Z4oeSNXHAb3aX62PMfcmybPyE\nUog8GpaweiCOvCDP48ojIwmd6a8BW/j9kIXLv+W0TWemAQXhU7KswcRbTTkA7JVlqabc/SaTLCq2\n9kHsTtcBTDNZtiLK791c9coM1HhNyc95knNqKah2gOWhMunPEUvWlIeIdMryeLc4hErzNY5IT6Gy\nJDHU+z6I6GQ/g6BFx/lBItc9ThqJHf6s/1yZ/3VBS6l+W/7/MJZipn/h2cbcBOEv7Od1p/iSTF0A\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1512 x^{2} y^{5}$$"
      ],
      "text/plain": [
       "      2  5\n",
       "1512⋅x ⋅y "
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(3*x**4*y**7, x, 2, y, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFEAAAAYBAMAAABw9gyhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1512 x^{2} y^{5}$$"
      ],
      "text/plain": [
       "      2  5\n",
       "1512⋅x ⋅y "
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(diff(3*x**4*y**7, x, x), y, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Às vezes, o SymPy pode retornar um resultado de uma forma pouco familiar. Se, por exemplo, você desejar usar o SymPy para verificar se você diferenciou algo corretamente, uma técnica que pode ser útil é subtrair o resultado do SymPy do seu resultado e verificar se a resposta é zero.\n",
    "\n",
    "Tomando o exemplo simples de uma função de base radial multiquádrica, $\\phi(r) = \\sqrt{r^2 + \\sigma^2}$ com $r = \\sqrt{x^2 + y^2}$ e $\\sigma$ uma constante, podemos verificar se a primeira derivada em $x$ é $\\frac{\\partial \\phi}{\\partial x} = \\frac{x}{\\sqrt{r^2 + \\sigma^ 2}}$.\n",
    "\n",
    "Neste exemplo, primeiro pedimos que o SymPy imprima a derivada. Veja que ela é impressa de forma diferente da nossa derivada de teste, mas a subtração verifica que elas são idênticas:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x/sqrt(x**2 + y**2 + σ**2)\n"
     ]
    }
   ],
   "source": [
    "r = sqrt(x**2 + y**2)\n",
    "sigma = Symbol('σ')\n",
    "def phi(x,y,sigma):\n",
    "    return sqrt(x**2 + y**2 + sigma**2)\n",
    "\n",
    "minha_dfdx= x/sqrt(r**2 + sigma**2)\n",
    "print(diff(phi(x, y, sigma), x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0\n"
     ]
    }
   ],
   "source": [
    "print(minha_dfdx - diff(phi(x, y, sigma), x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Aqui é trivial dizer que as expressões são idênticas sem a ajuda do SymPy, mas, em exemplos mais complicados, pode haver muitos outros termos, tornando-se cada vez mais difícil, demorado e propenso a erros tentar reorganizar nossa derivada de teste e a resposta do SymPy na mesma forma. São nesses casos que esta técnica de subtração é de maior utilidade. A integração usa uma sintaxe semelhante. Para o caso indefinido, especifique a função e a variável em relação à qual a integração é realizada:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAvBAMAAAAV7ydtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1klEQVQoFWNggICkywJQFgNvAe8GODuB/QOM\nzcCAEGdgkDsAF2eaBGeiqmH5BZOQO8DyHcbmcGB6CGQLKbuqKfAkJTswMDAmsDdxTIDIswmwfeQC\nioEAIwMn3CUMDPwKYEEwcf4AjM0loM/AJADhxS/Yz+AOlRBWErpeAFNEY/o/AlBsk1vaLpgZLBcY\n1sO8wPSLgb8BKsH6lEF+AkwRA0INMMDa4MIsGRfhbAaGvAAEhw8U+iDA4sDADYsh/o8INt8GBo4v\nECUM7AUM6xWgbIbYu1tBTABGZDQhWVh3aAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{x^{3}}{3}$$"
      ],
      "text/plain": [
       " 3\n",
       "x \n",
       "──\n",
       "3 "
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import integrate\n",
    "integrate(x**2, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACAAAAAYBAMAAABpfeIHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0klEQVQYGWNggIDcvQJQFoRic2CegCLA4cDz\nBVUggOUrigADA9dHNAEOBTSBXDQ++wU0gTCGGBQRnlmrNqII8P3//wEmIKTsqqYA5ghdYmB5yMCY\nwN7EAXF2wjoGpg8MbAJsH7kcQCpYFtxk4N7AwMjAuQGsAch6xMCvAGSDCZAY6xeG8weANJgACfBO\nYNADekpAn4FJAMRn4DZg2MzAEL9gP4M7mM/Au4H1OwODsJLQ9QKIAKOZLmoogU2ByEFISwb5AGQ+\nQzvDJhQ+Q5DuAQYGABRwKOS704Y5AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$x^{2} y$$"
      ],
      "text/plain": [
       " 2  \n",
       "x ⋅y"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x**2, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEUAAAAUBAMAAAApce1IAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nF1gAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$y \\sin{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "y⋅sin(x)"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(sin(x), y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAUBAMAAADVZMaCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$- \\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "-cos(x)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(sin(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACoAAAAaBAMAAAAzl9PFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAu0lEQVQoFWNgQAeL9ALQhYD8B2wFWEQvMH7E\nIsrAiU0tgyw2cxk3YjNAJFABTThY+TRD/X90cyfkMaGpA3ONsQkqPOPBIrzHyAqLKKoQ5zsgeMjA\n/x8KUGUZGLiUnqILAfl8DOlYRHkZ5AVAwozKJijBMR2sNj2AVQHMgBCMV0E0ayeDyAKIAJiUBhvA\n3WN8EEmQ0VUIxOPfgCTGwMD+/ztYVAFFFMrhAIoaYEgwX2AQQXEZRIWJzQEMpQBNOydIORes5gAA\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$e^{- \\frac{x^{2}}{2}}$$"
      ],
      "text/plain": [
       "   2 \n",
       " -x  \n",
       " ────\n",
       "  2  \n",
       "ℯ    "
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(-x*exp(-x**2/2), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos calcular integrais definidas fornecendo `integrate()` com uma tupla contendo a variável de interesse, os limites inferior e superior. Se várias variáveis forem especificadas, a integração múltipla é realizada. Quando o SymPy retorna um resultado na classe `Rational`, é possível avaliá-lo em uma representação de ponto flutuante com qualquer precisão desejada."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAHlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMAVO8Qq5l2zWYZcMvdAAAACXBIWXMAAA7EAAAOxAGV\nKw4bAAAAHUlEQVQIHWNgAANGZQYGk5DJQDYbqQSr03QPsBkAJYgIYEZbtZEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x*2, (x, 0, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAvBAMAAAAV7ydtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1klEQVQoFWNggICkywJQFgNvAe8GODuB/QOM\nzcCAEGdgkDsAF2eaBGeiqmH5BZOQO8DyHcbmcGB6CGQLKbuqKfAkJTswMDAmsDdxTIDIswmwfeQC\nioEAIwMn3CUMDPwKYEEwcf4AjM0loM/AJADhxS/Yz+AOlRBWErpeAFNEY/o/AlBsk1vaLpgZLBcY\n1sO8wPSLgb8BKsH6lEF+AkwRA0INMMDa4MIsGRfhbAaGvAAEhw8U+iDA4sDADYsh/o8INt8GBo4v\nECUM7AUM6xWgbIbYu1tBTABGZDQhWVh3aAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{x^{3}}{3}$$"
      ],
      "text/plain": [
       " 3\n",
       "x \n",
       "──\n",
       "3 "
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x**2, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAwBAMAAADnb5cjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA20lEQVQoFWNggAJuGANIrxeAcxjzEWwmeQT7\nFBI7AcFmd0CwhRgQ7LV371+Hm8ngDzJHSNlVTYGBfb8VAwNjAnsTxwSIPJsA20cuBwibkYFzA4QF\nIvkVEOzzB2BsLgF9Biaou+IX7Gdwh0oIKwldL4ApojH9HwGoYJPv7T1gU9YJMDAaMOQHAMPH1l6A\ngVmAge8BSEIfyGhgYP4DZXN+RLCBQhwfoeJAKj4BwW4GMUF6GRh4LyDYumAmWJzrAoMLiAdSc5aB\nQQXKZmlLu9wAZF9+P7mAE+hREBsBAEHnNkxqT1FdAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{x^{4}}{12}$$"
      ],
      "text/plain": [
       " 4\n",
       "x \n",
       "──\n",
       "12"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x**2, x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAAwBAMAAABtQA5iAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABOUlEQVQoFaXRu07DMBQG4D9WmpSECCt5gFYg\nsRXKwm2ALAg2EC9AJjYEz8DCSN+gfYFCBSNDM7O0gqFiiBBMTKiVym0KxzE4ilmQ6sH+8+Wc48HA\nz3J/gzovuYoyGMe6sIout38k0sUOdfGhSzt5HGi3Yze/y5/bmq/C7q7Dv4P5BBiRfVpuZB3RBdgQ\nsLg1ckIhZusBbodqMEW7WAaeMVMVSe4USmP0YiFyp+A1sECHwxfBuPgDt45rOg5aXWxnAK9T+qAU\nzPqDEynGcm0sU77TpMJaQ2WvADjDVRGwX4s1megz1dZwomn/azaTmzivdOiVAlhvSoL7EbAEvCoB\nI3kBNvM2IU2uC3Wcc9UmaugB3xVkc4ByXZckB1nD+rq0YYbKxOTpPlhRdg6PVlWJt/G1gmaafiop\nhm9/BlWskaX68AAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{x^{4} y}{12}$$"
      ],
      "text/plain": [
       " 4  \n",
       "x ⋅y\n",
       "────\n",
       " 12 "
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x**2, x, x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHaJmUSrVGYizbvd\n7zLJaKAlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAtUlEQVQYGWNgEFJ2ZQACBwa2AgYG9gYGzgUM\nDMwbGBgPMDCw/hZgbgDK5f+wAJIMzP8XgqiymL8BDAwcCQz6FxkYeCcw8H1nYJAHimeAeQw7gfoW\nMnAoAEWeOYONBrLIAf9B4ANROicbbweq43RgqG9gYGD6zsB/AOiIqwzyC4CiDGBBBgbGwyA2p40j\niGJgsAX6AQh4gX7gnMDADfQD/0cwxbuBgecn0H8NDPUKQAUhLlsZGACMyimEjS9YNgAAAABJRU5E\nrkJggg==\n",
      "text/latex": [
       "$$\\frac{8}{3}$$"
      ],
      "text/plain": [
       "8/3"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x**2, (x, 0, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAAqBAMAAACuFQ3dAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWaJMkS7\nIt2ZnNffAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAy0lEQVQYGWNggICZmQtADCEDBgbzDawFDAzM\nrv5A9lEGngMg8XwDBu6/IAYQANk8CmAWmM1fNS0azAOK21cwMAmAOCD2ZwZWZSib/wEDx28om6+A\ngeMHlM2igBDnAqpXgIozeDIYgcwJ0S9pYOAM9QIJ0xj8h4MPFNt0c+ZBmBm8AQzyDVAO20cGYEBA\nALc2g30BlA2k4GqAAf8SLsw7LxjOZmCYtAHBYQKHLJDPe4GB5yNUnP8zgs10gIEPFmEcDQzyCTC9\nW0NPg5gArp82VUe1SgEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{16}{3}$$"
      ],
      "text/plain": [
       "16/3"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x**2, (x, 0, 2), (x, 0, 2), (y, 0, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALMAAAAOBAMAAACWbjBrAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$2.6666666666666665$$"
      ],
      "text/plain": [
       "2.6666666666666665"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "float(integrate(x**2, (x, 0, 2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.numbers.Rational"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(integrate(x**2, (x, 0, 2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAOBAMAAABa9wqAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$2.66666666666667$$"
      ],
      "text/plain": [
       "2.66666666666667"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "resultado_racional=integrate(x**2, (x, 0, 2))\n",
    "resultado_racional.evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAgwAAAAOBAMAAACx/1keAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nOgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2.6666666666666666666666666666666666666666666666667$$"
      ],
      "text/plain": [
       "2.6666666666666666666666666666666666666666666666667"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "resultado_racional.evalf(50)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Equações diferenciais ordinárias\n",
    "\n",
    "O SymPy tem suporte incorporado para resolver vários tipos de equações diferenciais ordinárias através do comando `dsolve`. Precisamos configurar a EDO e passá-la como o primeiro argumento, `eq`. O segundo argumento é a função `f(x)` a resolver. Um terceiro argumento opcional, `hint`, influencia o método que `dsolve` usa: alguns métodos são mais adequados a certas classes de EDOs ou expressarão a solução de forma mais simples do que outros. \n",
    "\n",
    "Para configurar o solucionador de EDO, precisamos de alguma maneira, fazer referencia à função incógnita que estamos resolvendo, bem como às suas derivadas. As classes `Function` e ` Derivative` facilitam isso:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAZBAMAAAAPjOtOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$y{\\left (x \\right )} = C_{1} e^{- 5 x}$$"
      ],
      "text/plain": [
       "           -5⋅x\n",
       "y(x) = C₁⋅ℯ    "
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import Symbol, dsolve, Function, Derivative, Eq\n",
    "y = Function(\"y\")\n",
    "x = Symbol('x')\n",
    "y_ = Derivative(y(x), x)\n",
    "dsolve(y_ + 5*y(x), y(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Observe como `dsolve` introduziu uma constante de integração, `C1`. Esta função introduzirá tantas constantes quantas forem necessárias, e todas serão chamadas `Cn`, onde `n` é um número inteiro. Observe também que o primeiro argumento para `dsolve` é considerado igual a zero, a menos que usemos a função `Eq()` para especificar o contrário:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAZBAMAAAAPjOtOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$y{\\left (x \\right )} = C_{1} e^{- 5 x}$$"
      ],
      "text/plain": [
       "           -5⋅x\n",
       "y(x) = C₁⋅ℯ    "
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsolve(y_ + 5*y(x), y(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAZBAMAAAAPjOtOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$y{\\left (x \\right )} = C_{1} e^{- 5 x}$$"
      ],
      "text/plain": [
       "           -5⋅x\n",
       "y(x) = C₁⋅ℯ    "
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsolve(Eq(y_ + 5*y(x), 0), y(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALoAAAAqBAMAAAD2RlyBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n5tCU/wepFtoiB0TbLQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$y{\\left (x \\right )} = \\frac{C_{1}}{5} e^{- 5 x} + \\frac{12}{5}$$"
      ],
      "text/plain": [
       "           -5⋅x     \n",
       "       C₁⋅ℯ       12\n",
       "y(x) = ──────── + ──\n",
       "          5       5 "
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsolve(Eq(y_ + 5*y(x), 12), y(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Os resultados de `dsolve` são uma instância da classe `Equality`. Isto tem consequências quando desejamos avaliar numericamente a função e usar o resultado em outro lugar (e.g. se quisermos plotar *y*(*x*) por *x*), porque mesmo depois de usar `subs()` e `evalf()`, ainda temos uma `Equality`, e não um escalar. A maneira de avaliar a função para um número é através do atributo `rhs` da `Equality`. Note que, aqui, usamos `z` para armazenar a `Equality` retornada por `dsolve`, ainda que seja uma expressão para uma função chamada `y(x)`, enfatizando a distinção entre `Equality` e os dados que ela contém."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAZBAMAAAAPjOtOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$y{\\left (x \\right )} = C_{1} e^{- 5 x}$$"
      ],
      "text/plain": [
       "           -5⋅x\n",
       "y(x) = C₁⋅ℯ    "
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = dsolve(y_ + 5*y(x), y(x))\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.relational.Equality"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAD8AAAAXBAMAAABOjoq3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHa7q2Yiie9Umd3N\nRDIfxLosAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABIklEQVQoFWNgwA74/FIDsMtARXn/f8Erz8A7\nT4CAggv45Rl4F2kJMBStU3mASx3bBVYDJgEfbpBBjMourmwFGCr5vvExHAWLagQwcK4IgCuYbAwE\nAiwCjH8ZGL6BRIU3Aon1CSAmEuBIYPvIwPeBDyh0XgBI3ECSAzPZBdgX5LNvYGdg4PoJEniFroCh\nfDmDiG5RIQND/gaQnBhEgbjSMgyV9QoQIb4DDAy82xjSMRTsfwAW4lsCNCljAuM6DAX9IDcyCDBw\nAhV4BKmCeSiK9oN4nA/ACrBGIDgEuIBqgCZ8R9EK5eQXABlA34AUtDMwMGGo4fzKwDALaDNIgQ4D\nI6Y3GaRMvC4AtYEUMC0KxTAALgBSgBdwNuCVZmBb/UUBqwoAhPA7g2Zes5kAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$C_{1} e^{- 5 x}$$"
      ],
      "text/plain": [
       "    -5⋅x\n",
       "C₁⋅ℯ    "
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z.rhs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGAAAAAqBAMAAAC+dUMBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nA5bvfUy3GZw9AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$y{\\left (3 \\right )} = \\frac{2}{e^{15}}$$"
      ],
      "text/plain": [
       "          -15\n",
       "y(3) = 2⋅ℯ   "
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C1=Symbol('C1')\n",
    "y3 = z.subs({C1:2, x:3})\n",
    "y3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANwAAAAZBAMAAACoUV21AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
      "text/latex": [
       "$$y{\\left (3 \\right )} = 6.11804641 \\cdot 10^{-7}$$"
      ],
      "text/plain": [
       "y(3) = 6.11804641e-7"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y3.evalf(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABwAAAAqBAMAAABSACYXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5klEQVQoFWNgAAFGJWMHMANChDGw/0DipjIw\nzETizmJgyHdA8PcHoHCBEusDELIMDBzfkXkMXAkoXCUUHo8CCvcSA6MAQoBFgYEHiStVXpSFkGTY\n////LyTuADD/owCiHMCygYElRe4BVC2L2gIGjv/f4Dp5gdwriLAEcRvgkgwgrs4huDSQy9jAZwCT\nB3IZGFg+InHZAlhBkcpac0gApJdTgPErkNslwFjAwDuBgT2AXQGYDuYyBDkwan07wFCuDpRkm3fm\nKZCCAf4EGAtM8xegcDmB3AMIEZ4NDEECCC7DoXMPEDwAq0s8vgnwWrMAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{2}{e^{15}}$$"
      ],
      "text/plain": [
       "   -15\n",
       "2⋅ℯ   "
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y3.rhs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANwAAAAZBAMAAACoUV21AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHaZIu+JVM27RDKr\nZt2dj8xZAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADXklEQVRIDb1WTUhUURT+njO+cX59BlYL0UGp\nVaAghZbQQL8uihEK0YIeI4gS1XPRIiocqEW/MFItBMPZBOGiEYK2WmJBKT7CokWhUFBBZFrGaNp0\n7t9ramapc8BzvnPud+/nPe/eq0DBTO8pMwomBgQymaVCyg0N9RVQzgNPsoBywEZHzW0LOO1U1gE8\ndtZsBtyd3RYCEaekgB47qaCIWnVHEtqO18OU+uvIdW5l8KEY/ddLGi/6E87YZWAPPKvQ4k5JAu0a\nGjj0tFBgzgU8Q4mlfaA8REtsTrriBD/Tz1mDnDLG5bRz/WQG9Do1EiRUa+E3EFMlFb116Ge4omle\nuovAQbwCuqi6heRuwUu+qA3a2JShpkmupPFqiakGQxHgXlJLA+WqpGKKhrgVkRyY6wYqMQD0GnC/\nScD3gw8fukqhNkuOcwWNE1Bsiwgc5oCaiZKkqsl4R+WOXKoPp7ACpCzoJQl4o5xh5pUTNLFEKAlo\nNWHsM2p4odJEVoMFB+lHNUQjc+QCmde29p3kwpgmudLrsZ1AMJJPTtL4fO50TxS1uMvw/vfUimCc\nwa4rzC4R0tI2WlnprxzOpA33T2DE1MIkV3UTxRY2IJ+coPHpwh2gU9WPjzzR2+nozXLoOC1jYDv/\nIs7uPM/n2t20uxFTB5Obh6sFZrYc57PfT9CcxQCjOIwFkA6zowb83wR0PJ3WmSTLHLm3KFos4808\nz+RKZxFc9gxnybEjJybkNBOpiP8Xl9sLDFq5cp9IzpazheYR2phFZ6C3x2RyoTiCq/Q+ZTXzi5Jj\nJ6rX4pl0DSiK8mZS10jON8vqp78y47PG/9+dRvsP2HQXZzZNTEwdDweitLsLExOLT/JcBEYzpBIP\nA/CO8qPSBsxF5FHJIozQt+Op00zanTdC95c9NsUJ+OnbRQnS/Nx7J2l8Ae4akbLRQfAdPAv0QpkE\nsy1gaq2oimd9u/sGXiJkaceIVpoAnqLcIrhEPznXXNJoSFpFbHcEDyjx1Tcm6ZoPqwEVO5qGERqF\nazw9KZx/jD3RsW3E1AdXbOj1L4jbnZlE04kbtpomuILm1BigprBHTJh4X1S25jEQ15bpctephXOe\naDWwNjFke8O00m25Wu4foLWRUav4qncx2Cxzb0SC9Q0F+ecB+APkvQFlvNHDQQAAAABJRU5ErkJg\ngg==\n",
      "text/latex": [
       "$$y{\\left (3 \\right )} = 6.11804641 \\cdot 10^{-7}$$"
      ],
      "text/plain": [
       "y(3) = 6.11804641e-7"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y3.evalf(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABwAAAAqBAMAAABSACYXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5klEQVQoFWNgAAFGJWMHMANChDGw/0DipjIw\nzETizmJgyHdA8PcHoHCBEusDELIMDBzfkXkMXAkoXCUUHo8CCvcSA6MAQoBFgYEHiStVXpSFkGTY\n////LyTuADD/owCiHMCygYElRe4BVC2L2gIGjv/f4Dp5gdwriLAEcRvgkgwgrs4huDSQy9jAZwCT\nB3IZGFg+InHZAlhBkcpac0gApJdTgPErkNslwFjAwDuBgT2AXQGYDuYyBDkwan07wFCuDpRkm3fm\nKZCCAf4EGAtM8xegcDmB3AMIEZ4NDEECCC7DoXMPEDwAq0s8vgnwWrMAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{2}{e^{15}}$$"
      ],
      "text/plain": [
       "   -15\n",
       "2⋅ℯ   "
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y3.rhs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJsAAAAVBAMAAACj9YEXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n1dj9sAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$6.11804641 \\cdot 10^{-7}$$"
      ],
      "text/plain": [
       "6.118046410e-7"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y3.rhs.evalf(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKUAAAAVBAMAAADVzzOcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$4.122307245 \\cdot 10^{-9}$$"
      ],
      "text/plain": [
       "4.122307245e-9"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z.rhs.subs({C1:2, x:4}).evalf(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKsAAAAVBAMAAADLBgMvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nqEEXn9s+8W+fvwCUFNjcx6UifwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2.777588773 \\cdot 10^{-11}$$"
      ],
      "text/plain": [
       "2.777588773e-11"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z.rhs.subs({C1:2, x:5}).evalf(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.numbers.Float"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(z.rhs.subs({C1:2, x:5}).evalf(10))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Às vezes, `dsolve` pode retornar uma solução muito geral. Um exemplo é quando existe a possibilidade de que alguns coeficientes sejam complexos. Se soubermos que, por exemplo, eles são sempre reais e positivos, podemos passar esta informação para `dsolve` e evitar que a solução se torne desnecessariamente complicada:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAO8AAAAcBAMAAAB7WPU9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
      "text/latex": [
       "$$f{\\left (x \\right )} = C_{1} e^{- i a^{2} x} + C_{2} e^{i a^{2} x}$$"
      ],
      "text/plain": [
       "               2            2  \n",
       "           -ⅈ⋅a ⋅x       ⅈ⋅a ⋅x\n",
       "f(x) = C₁⋅ℯ        + C₂⋅ℯ      "
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *\n",
    "a, x = symbols('a,x')\n",
    "f = Function('f')\n",
    "dsolve(Derivative(f(x), x, 2) + a**4*f(x), f(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAATQAAAAcBAMAAAAdJ15DAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nS/n3BBL4Lfj/tX8AlPhFKQW/CckAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$f{\\left (x \\right )} = C_{1} \\sin{\\left (a^{2} x \\right )} + C_{2} \\cos{\\left (a^{2} x \\right )}$$"
      ],
      "text/plain": [
       "             ⎛ 2  ⎞         ⎛ 2  ⎞\n",
       "f(x) = C₁⋅sin⎝a ⋅x⎠ + C₂⋅cos⎝a ⋅x⎠"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a=Symbol('a',real=True,positive=True)\n",
    "dsolve(Derivative(f(x), x, 2)+a**4*f(x), f(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Expansões em série e plotagens\n",
    "\n",
    "É possível expandir muitas expressões do SymPy em série de Taylor. O método `series` torna isso direto. No mínimo, devemos especificar a expressão e a variável a ser expandida. Opcionalmente, também podemos especificar o ponto em torno do qual expandir, o número máximo de termos e a direção da expansão (tente `help(Basic.series)` para mais informações)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANEAAAAwBAMAAAB9DIL7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nJGzV0PoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x - \\frac{x^{3}}{6} + \\frac{x^{5}}{120} + \\mathcal{O}\\left(x^{6}\\right)$$"
      ],
      "text/plain": [
       "     3     5        \n",
       "    x     x     ⎛ 6⎞\n",
       "x - ── + ─── + O⎝x ⎠\n",
       "    6    120        "
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *\n",
    "x = Symbol('x')\n",
    "sin(x).series(x, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANEAAAAwBAMAAAB9DIL7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nJGzV0PoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x - \\frac{x^{3}}{6} + \\frac{x^{5}}{120} + \\mathcal{O}\\left(x^{6}\\right)$$"
      ],
      "text/plain": [
       "     3     5        \n",
       "    x     x     ⎛ 6⎞\n",
       "x - ── + ─── + O⎝x ⎠\n",
       "    6    120        "
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(sin(x), x, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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vtF3w0kCRNHZcu5Zj7hmWwbSb0JQSm20RrCD5xL4PzZs36Z3CyVks2g9CbcVW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\njMxvnOPfUtP/B7GMp8/WewdbAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$1.11729533119247 - 0.438791280945186 \\left(x - 0.5\\right)^{2} + 0.0799042564340338 \\left(x - 0.5\\right)^{3} + 0.0365659400787655 \\left(x - 0.5\\right)^{4} - 0.00399521282170169 \\left(x - 0.5\\right)^{5} - 0.00121886466929218 \\left(x - 0.5\\right)^{6} + 9.51241148024212 \\cdot 10^{-5} \\left(x - 0.5\\right)^{7} + 2.17654405230747 \\cdot 10^{-5} \\left(x - 0.5\\right)^{8} - 1.32116826114474 \\cdot 10^{-6} \\left(x - 0.5\\right)^{9} - 0.479425538604203 x + \\mathcal{O}\\left(\\left(x - 0.5\\right)^{10}; x\\rightarrow0.5\\right)$$"
      ],
      "text/plain": [
       "                                              2                               \n",
       "1.11729533119247 - 0.438791280945186⋅(x - 0.5)  + 0.0799042564340338⋅(x - 0.5)\n",
       "\n",
       "3                               4                                5            \n",
       "  + 0.0365659400787655⋅(x - 0.5)  - 0.00399521282170169⋅(x - 0.5)  - 0.0012188\n",
       "\n",
       "                    6                                7                        \n",
       "6466929218⋅(x - 0.5)  + 9.51241148024212e-5⋅(x - 0.5)  + 2.17654405230747e-5⋅(\n",
       "\n",
       "        8                                9                          ⎛         \n",
       "x - 0.5)  - 1.32116826114474e-6⋅(x - 0.5)  - 0.479425538604203⋅x + O⎝(x - 0.5)\n",
       "\n",
       "10         ⎞\n",
       "  ; x → 0.5⎠"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cos(x).series(x, 0.5, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Em alguns casos, especialmente para avaliação numérica e  plotagem dos resultados, é necessário remover o termo final `O (n)`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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mTF4MVObJ1bWCpxnzEAVFQhkaAk+MI8Ds5IIuFocWODtjQVekCHhKi7UIEd4StiiXy6TQ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\nZZkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- 0.479425538604203 x - 1.32116826114474 \\cdot 10^{-6} \\left(x - 0.5\\right)^{9} + 2.17654405230747 \\cdot 10^{-5} \\left(x - 0.5\\right)^{8} + 9.51241148024212 \\cdot 10^{-5} \\left(x - 0.5\\right)^{7} - 0.00121886466929218 \\left(x - 0.5\\right)^{6} - 0.00399521282170169 \\left(x - 0.5\\right)^{5} + 0.0365659400787655 \\left(x - 0.5\\right)^{4} + 0.0799042564340338 \\left(x - 0.5\\right)^{3} - 0.438791280945186 \\left(x - 0.5\\right)^{2} + 1.11729533119247$$"
      ],
      "text/plain": [
       "                                                    9                         \n",
       "-0.479425538604203⋅x - 1.32116826114474e-6⋅(x - 0.5)  + 2.17654405230747e-5⋅(x\n",
       "\n",
       "       8                                7                                6    \n",
       " - 0.5)  + 9.51241148024212e-5⋅(x - 0.5)  - 0.00121886466929218⋅(x - 0.5)  - 0\n",
       "\n",
       "                            5                               4                 \n",
       ".00399521282170169⋅(x - 0.5)  + 0.0365659400787655⋅(x - 0.5)  + 0.079904256434\n",
       "\n",
       "              3                              2                   \n",
       "0338⋅(x - 0.5)  - 0.438791280945186⋅(x - 0.5)  + 1.11729533119247"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cos(x).series(x, 0.5, 10).removeO()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "O SymPy fornece duas funções de plotagem, `plot()`, do módulo `sympy.plotting`, e `plot` do módulo `sympy.mpmath.visualization`. No momento da escrita, essas funções careciam de algumas capacidades de plotagem, o que significa que não são adequadas para a maioria de nossas necessidades. Ainda assim, caso você deseje utilizá-las, a função `help()` delas é útil. \n",
    "\n",
    "Para a maioria dos nossos propósitos, Matplotlib deve ser a ferramenta de plotagem a ser escolhida. Aqui, fornecemos apenas um exemplo de como plotar os resultados de uma computação SymPy."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1140d6c50>"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1140d6c50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sympy import sin,series,Symbol\n",
    "import pylab\n",
    "x = Symbol('x')\n",
    "s10 = sin(x).series(x,0,10).removeO()\n",
    "s20 = sin(x).series(x,0,20).removeO()\n",
    "s = sin(x)\n",
    "xx = []\n",
    "y10 = []\n",
    "y20 = []\n",
    "y = []\n",
    "for i in range(1000):\n",
    "  xx.append(i / 100.0)\n",
    "  y10.append(float(s10.subs({x:i/100.0})))\n",
    "  y20.append(float(s20.subs({x:i/100.0})))\n",
    "  y.append(float(s.subs({x:i/100.0})))\n",
    "\n",
    "pylab.figure()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x1143f4a58>"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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DXZ5y/fMgXlMcju9cqXlIcRBuYi/IZf3mX1i09VcWp29iPXnkFzb/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\nIU7z/+dIsjdNDUrMAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11404f828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pylab.plot(xx, y10, label='O(10)')\n",
    "pylab.plot(xx, y20, label='O(20)')\n",
    "pylab.plot(xx, y, label='sin(x)')\n",
    "\n",
    "pylab.axis([0, 10, -4, 4])\n",
    "pylab.xlabel('x')\n",
    "pylab.ylabel('f(x)')\n",
    "\n",
    "pylab.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Equações lineares e inversão de matrizes\n",
    "\n",
    "O SymPy possui uma classe `Matrix` e funções associadas que permitem a solução simbólica de sistemas de equações lineares (e, claro, podemos obter respostas numéricas com `subs()` e `evalf()`). Considerararemos o exemplo do seguinte par de equações lineares simples:\n",
    "\n",
    "$$ \n",
    "\\begin{aligned}\n",
    "3x + 7y & = 12z \\\\\n",
    "4x - 2y & = 5z \n",
    "\\end {aligned} \n",
    "$$\n",
    "\n",
    "Podemos escrever este sistema na forma $\\textbf{A} \\textbf{x} = \\textbf{b}$ (multiplique $\\textbf{A}$ por $\\textbf{x}$ caso queira verificar se as equações originais são recuperadas), onde\n",
    "\n",
    "$${\\bf A} = \n",
    "\\left(\n",
    "\\begin{array}{cc}\n",
    "3 & 7 \\\\\n",
    "4 & -2 \n",
    "\\end{array} \n",
    "\\right), \n",
    "\\qquad\n",
    "\\textbf{x} = \n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    "x \\\\\n",
    "y \n",
    "\\end{array} \n",
    "\\right), \n",
    "\\qquad\n",
    "\\textbf{b} = \n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    "12z \\\\\n",
    "5z \n",
    "\\end{array} \n",
    "\\right).$$\n",
    "\n",
    "Aqui, incluímos um símbolo, $z$, no lado direito para demonstrar que os símbolos serão propagados na solução. Em muitos casos, teríamos $z = 1$, mas ainda pode haver benefício ao se usar SymPy como um solucionador numérico, mesmo quando a solução não contenha símbolos, devido à sua capacidade de retornar frações exatas em vez de `floats` aproximados.\n",
    "\n",
    "Uma estratégia para resolver o sistema para $\\textbf{x}$ é inverter a matriz $\\textbf{A}$ e pré-multiplicar a equação pela matriz inversa, i.e. $\\textbf{A}^{-1} \\textbf{A} \\textbf{x} = \\textbf{x} = \\textbf{A}^{-1} \\textbf{b}$. A classe `Matrix` do SymPy possui um método `inv()` que nos permite encontrar a inversa e `*` realiza a multiplicação da matriz para nós, quando apropriado:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAAAyBAMAAAATs7BgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}3 & 7\\\\4 & -2\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡3  7 ⎤\n",
       "⎢     ⎥\n",
       "⎣4  -2⎦"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import symbols,Matrix\n",
    "x, y, z = symbols('x,y,z')\n",
    "A = Matrix(([3, 7], [4, -2]))\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFwAAAA0CAMAAAAwlq9dAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{1}{17} & \\frac{7}{34}\\\\\\frac{2}{17} & - \\frac{3}{34}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1/17  7/34 ⎤\n",
       "⎢           ⎥\n",
       "⎣2/17  -3/34⎦"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.inv()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACsAAAAyBAMAAAA3qj+gAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}12 z\\\\5 z\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡12⋅z⎤\n",
       "⎢    ⎥\n",
       "⎣5⋅z ⎦"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = Matrix(( 12*z,5*z  ))\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACoAAAA0CAMAAADLwVqCAAAASFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{59 z}{34}\\\\\\frac{33 z}{34}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡59⋅z⎤\n",
       "⎢────⎥\n",
       "⎢ 34 ⎥\n",
       "⎢    ⎥\n",
       "⎢33⋅z⎥\n",
       "⎢────⎥\n",
       "⎣ 34 ⎦"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = A.inv()*b\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADsAAAAyBAMAAAAQBL5IAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n978AhhSn6fp3bhEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}5.726\\\\3.203\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡5.726⎤\n",
       "⎢     ⎥\n",
       "⎣3.203⎦"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.subs({z:3.3}).evalf(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.matrices.dense.MutableDenseMatrix"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Um método alternativo para resolver o mesmo problema é construir o sistema como uma matriz aumentada. Essa é a forma obtida dispondo juntas as colunas (no nosso exemplo) de $\\textbf{A}$ e o vetor $\\textbf{b}$. A matriz aumentada é [1]:\n",
    "\n",
    "$$ \n",
    "(\\textbf{A} \\, | \\, \\textbf{b} ) = \n",
    "\\left( \n",
    "\\begin{array}{cc|c}\n",
    "3 & 7 & 12z \\\\\n",
    "4 & -2 & 5z \n",
    "\\end{array} \n",
    "\\right), \n",
    "$$ \n",
    "e, como antes, construímos isto como um objeto SymPy `Matrix`, mas, neste caso, passamos para a função `solve_linear_system()`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHoAAAAyBAMAAABhfNLWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}3 & 7 & 12 z\\\\4 & -2 & 5 z\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡3  7   12⋅z⎤\n",
       "⎢           ⎥\n",
       "⎣4  -2  5⋅z ⎦"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import Matrix, symbols, solve_linear_system\n",
    "x, y, z = symbols('x,y,z')\n",
    "system = Matrix(([3, 7, 12*z],[4, -2, 5*z]))\n",
    "system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALsAAAAyBAMAAAD2AbdJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$\\left \\{ x : \\frac{59 z}{34}, \\quad y : \\frac{33 z}{34}\\right \\}$$"
      ],
      "text/plain": [
       "⎧   59⋅z     33⋅z⎫\n",
       "⎨x: ────, y: ────⎬\n",
       "⎩    34       34 ⎭"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol = solve_linear_system(system,x,y)\n",
    "sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dict"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(sol)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x = 5.726\n",
      "y = 3.203\n"
     ]
    }
   ],
   "source": [
    "for k in sol.keys():\n",
    "    print(k,'=',sol[k].subs({z:3.3}).evalf(4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Uma terceira opção é o método `solve()`, cujos argumentos incluem as equações simbólicas individuais, em vez de matrizes. Como `dsolve()`, `solve()` espera ou expressões que ela assumirá como iguais a zero, ou objetos `Equality`, que podemos criar convenientemente com `Eq()`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALsAAAAyBAMAAAD2AbdJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$\\left \\{ x : \\frac{59 z}{34}, \\quad y : \\frac{33 z}{34}\\right \\}$$"
      ],
      "text/plain": [
       "⎧   59⋅z     33⋅z⎫\n",
       "⎨x: ────, y: ────⎬\n",
       "⎩    34       34 ⎭"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import symbols,solve,Eq\n",
    "x, y, z = symbols('x,y,z')\n",
    "solve((Eq(3*x+7*y,12*z), Eq(4*x-2*y,5*z)), x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALsAAAAyBAMAAAD2AbdJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$\\left \\{ x : \\frac{59 z}{34}, \\quad y : \\frac{33 z}{34}\\right \\}$$"
      ],
      "text/plain": [
       "⎧   59⋅z     33⋅z⎫\n",
       "⎨x: ────, y: ────⎬\n",
       "⎩    34       34 ⎭"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve((3*x+7*y-12*z, 4*x-2*y-5*z), x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para obter mais informações, consulte `help(solve)` e `help(solve_linear_system)`.\n",
    "\n",
    "### Equações não-lineares\n",
    "\n",
    "Vamos resolver uma equação simples como $x = x^2$. Existem duas soluções óbvias: $x = 0$ e $x = 1$. Como podemos pedir ao SymPy para calcular isso para nós?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy\n",
    "x, y, z = sympy.symbols('x, y, z')        # cria alguns símbolos\n",
    "eq = x - x ** 2                           # define a equação"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADoAAAAUBAMAAAAuMzNdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdt3NMolEEJlmVCLv\nu6sHwGgPAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAuklEQVQoFWMQMmHABSarMYQxsLdOdcCuIAwo\ny7yAXRtTdk0BA1AujGEOA0Mwuiz7i/dQ2XsMDOuBTDQgD5X9x8CQvwBNjoEBKsv+AygrgEuW+ysD\ng38ATlmgXtyy+E1mALpqPU5XMdxlYJiP20fA0GjGcBTMRwxMC9iPMNQboCmAhQZ7oJwDA5M6qmzf\nebsJ4HCGCK9GlQXzQLEAAZjhAYkjsCy7AFQVMgXXy4ksCmOHMQipwNgYtJAaAME1L4vUTMlrAAAA\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ 0, \\quad 1\\right ]$$"
      ],
      "text/plain": [
       "[0, 1]"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(eq, x)                        # resolve eq = 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A função `solve()` espera uma expressão que deve ser resolvida com o zero isolado. Para o nosso exemplo, reescrevemos $x = x^2$ como $x - x^2 = 0$ e, em seguida, a passamos para a função `solve()`.\n",
    "\n",
    "Vamos repetir o mesmo para a equação $x = x^3$ e resolver"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHMAAAAUBAMAAABMnB83AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdt3NMolEEFTvq5lm\nIrvActx7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA8UlEQVQ4EWMQMmEgB/AYJTCE4dfIvvWoA1YV\nrKhaOwswVDE3sOtiCHJMZ2BA0cq+ej2m1pMMDMHoWqt2fUfTysAgj6n1FQNDP4YoI1FafzEw5Deg\nW0uUVvbfQK0CZGnl/crA4B9AnlagrXi0shspAYGyA7ZgosDBDMBg6m8gy8EM7xgYzpMZOcAksRnd\nUgbskcP+GVUhUwP7FIZ6A1RBTK275lsdYFBCdR57oJQDA5M6ilbulZ/XoKZhiDTHBRRlEE4vphhK\n8odIs2KqYmAIwBTEorUQUxUDuwCmIBatBzBVMfBgEQNqFVLBIk5YiFEpAQAQz0QLz9I55QAAAABJ\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ -1, \\quad 0, \\quad 1\\right ]$$"
      ],
      "text/plain": [
       "[-1, 0, 1]"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq = x - x ** 3                           # define a equação\n",
    "sympy.solve(eq, x)                        # resolve eq = 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Saída: interface com LaTeX e impressão elegante\n",
    "\n",
    "Como é o caso de muitos sistemas algébricos computacionais, o SymPy tem a capacidade de formatar sua saída como código LaTeX para facilitar sua inclusão em documentos. \n",
    "\n",
    "No início deste capítulo, chamamos:\n",
    "\n",
    "``` python\n",
    "sympy.init_printing()\n",
    "```\n",
    "\n",
    "O SymPy detectou que estava em Jupyter e habilitou a saída em LaTeX. O Jupyter Notebook suporta (um pouco de) LaTeX e isso é o que nos proporciona a saída elegantemente formatada vista acima. Também podemos ver a saída em texto simples a partir do SymPy, e o código LaTeX puro que pode ser gerado:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y**2/x**3 - y/x**2 + 1/x + O(y**3)\n"
     ]
    }
   ],
   "source": [
    "print(series(1/(x+y), y, 0, 3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\\frac{y^{2}}{x^{3}} - \\frac{y}{x^{2}} + \\frac{1}{x} + \\mathcal{O}\\left(y^{3}\\right)\n"
     ]
    }
   ],
   "source": [
    "print(latex(series(1/(x+y), y, 0, 3)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "$\\frac{y^{2}}{x^{3}} - \\frac{y}{x^{2}} + 1 / x + \\mathcal{O}\\left(y^{3}\\right)$\n"
     ]
    }
   ],
   "source": [
    "print(latex(series(1/(x+y), y, 0, 3), mode='inline'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Esteja ciente de que em seu modo padrão, o código de saída produzido por `latex()` requer o pacote `amsmath`, a ser carregado através do comando `\\usepackage amsmath}` no preâmbulo do documento. \n",
    "\n",
    "O SymPy também suporta uma rotina de \"impressão elegante\" com a função `pprint()` (abreviatura de \"pretty print\"), que produz saída de texto com melhor formatação do que a rotina de impressão padrão. Observe os recursos de notação, tais como subíndices para elementos de um `array` cujos nomes são da forma `T_n`, a constante italicizada $e$, pontos centralizados verticalmente para a multiplicação, bordas de matrizes e frações.\n",
    "\n",
    "Finalmente, o SymPy oferece a função `preview()`, que exibe a saída renderizada na tela (verifique `help(preview)` para obter detalhes)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Geração automática de código em C \n",
    "\n",
    "Um ponto forte de muitas bibliotecas simbólicas é que elas podem converter as expressões simbólicas em código C (ou outro código), o qual, posteriormente, pode ser compilado para execução em alta velocidade. Aqui está um exemplo que demonstra isso:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANEAAAAwBAMAAAB9DIL7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nJGzV0PoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x - \\frac{x^{3}}{6} + \\frac{x^{5}}{120} + \\mathcal{O}\\left(x^{6}\\right)$$"
      ],
      "text/plain": [
       "     3     5        \n",
       "    x     x     ⎛ 6⎞\n",
       "x - ── + ─── + O⎝x ⎠\n",
       "    6    120        "
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *                                                                                    \n",
    "from sympy.utilities.codegen import codegen                                                                            \n",
    "x = Symbol('x')                                                                                                          \n",
    "sin(x).series(x, 0, 6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "/******************************************************************************\n",
      " *                       Code generated with sympy 1.0                        *\n",
      " *                                                                            *\n",
      " *              See http://www.sympy.org/ for more information.               *\n",
      " *                                                                            *\n",
      " *                       This file is part of 'project'                       *\n",
      " ******************************************************************************/\n",
      "#include \"taylor_sine.h\"\n",
      "#include <math.h>\n",
      "\n",
      "double taylor_sine(double x) {\n",
      "\n",
      "   double taylor_sine_result;\n",
      "   taylor_sine_result = x - 1.0L/6.0L*pow(x, 3) + (1.0L/120.0L)*pow(x, 5) + O(x**6);\n",
      "   return taylor_sine_result;\n",
      "\n",
      "}\n",
      "\n"
     ]
    }
   ],
   "source": [
    "print(codegen((\"taylor_sine\",sin(x).series(x,0,6)), language='C')[0][1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ferramentas relacionadas\n",
    "-------------\n",
    "\n",
    "Vale a pena conferir a iniciativa SAGE (<http://www.sagemath.org/>), que está tentando \"criar uma alternativa de fonte aberta livre e viável para Magma, Maple, Mathematica e Matlab\" e inclui a biblioteca SymPy e muitas outras. Suas capacidades simbólicas são mais poderosas do que as do SymPy e o SAGE, à exceção dos recursos do SymPy, já cobrirá muitas das necessidades emergentes nos campos das ciências e engenharias. O SAGE inclui o sistema algébrico computacional Maxima, que também está disponível de forma autônoma em <http://maxima.sourceforge.net/>."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
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 },
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