SciPy_SymPy/tutorial_exercises/Solvers Solutions.ipynb

{
 "metadata": {
  "name": "Solvers Solutions"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Solvers"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Boilerplate to make the doctester work. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import sys\n",
      "import os\n",
      "sys.path.insert(1, os.path.join(os.path.pardir, \"ipython_doctester\"))\n",
      "from sympy import *\n",
      "from ipython_doctester import test\n",
      "# Work around a bug in IPython. This will disable the ability to paste things with >>>\n",
      "def notransform(line): return line\n",
      "from IPython.core import inputsplitter\n",
      "inputsplitter.transform_classic_prompt = notransform\n",
      "init_printing()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For each exercise, fill in the function according to its docstring. Execute the cell to see if you did it right. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a, b, c, d, x, y, z, t = symbols('a b c d x y z t')\n",
      "f, g, h = symbols('f g h', cls=Function)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Algebraic Equations"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Write a function that computes the [quadratic equation](http://en.wikipedia.org/wiki/Quadratic_equation)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def quadratic():\n",
      "    return solve(a*x**2 + b*x + c, x)\n",
      "quadratic()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}\\frac{- b + \\sqrt{- 4 a c + b^{2}}}{2 a}, & - \\frac{b + \\sqrt{- 4 a c + b^{2}}}{2 a}\\end{bmatrix}$$"
       ],
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "\u23a1        _____________   \u239b       _____________\u239e\u23a4\n",
        "\u23a2       \u2571           2    \u239c      \u2571           2 \u239f\u23a5\n",
        "\u23a2-b + \u2572\u2571  -4\u22c5a\u22c5c + b    -\u239db + \u2572\u2571  -4\u22c5a\u22c5c + b  \u23a0\u23a5\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a3         2\u22c5a                     2\u22c5a          \u23a6"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Write a function that computes the general solution to the cubic $x^3 + ax^2 + bx + c$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def cubic():\n",
      "    return solve(x**3 + a*x**2 + b*x + c, x)\n",
      "cubic()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}- \\frac{1}{3} a + \\frac{- \\frac{1}{9} a^{2} + \\frac{1}{3} b}{\\sqrt[3]{\\frac{1}{27} a^{3} - \\frac{1}{6} a b + \\frac{1}{2} c + \\sqrt{\\left(- \\frac{1}{9} a^{2} + \\frac{1}{3} b\\right)^{3} + \\frac{1}{4} \\left(\\frac{2}{27} a^{3} - \\frac{1}{3} a b + c\\right)^{2}}}} - \\sqrt[3]{\\frac{1}{27} a^{3} - \\frac{1}{6} a b + \\frac{1}{2} c + \\sqrt{\\left(- \\frac{1}{9} a^{2} + \\frac{1}{3} b\\right)^{3} + \\frac{1}{4} \\left(\\frac{2}{27} a^{3} - \\frac{1}{3} a b + c\\right)^{2}}}, & - \\frac{1}{3} a + \\frac{- \\frac{1}{9} a^{2} + \\frac{1}{3} b}{\\left(- \\frac{1}{2} - \\frac{1}{2} \\sqrt{3} \\mathbf{\\imath}\\right) \\sqrt[3]{\\frac{1}{27} a^{3} - \\frac{1}{6} a b + \\frac{1}{2} c + \\sqrt{\\left(- \\frac{1}{9} a^{2} + \\frac{1}{3} b\\right)^{3} + \\frac{1}{4} \\left(\\frac{2}{27} a^{3} - \\frac{1}{3} a b + c\\right)^{2}}}} - \\left(- \\frac{1}{2} - \\frac{1}{2} \\sqrt{3} \\mathbf{\\imath}\\right) \\sqrt[3]{\\frac{1}{27} a^{3} - \\frac{1}{6} a b + \\frac{1}{2} c + \\sqrt{\\left(- \\frac{1}{9} a^{2} + \\frac{1}{3} b\\right)^{3} + \\frac{1}{4} \\left(\\frac{2}{27} a^{3} - \\frac{1}{3} a b + c\\right)^{2}}}, & - \\frac{1}{3} a + \\frac{- \\frac{1}{9} a^{2} + \\frac{1}{3} b}{\\left(- \\frac{1}{2} + \\frac{1}{2} \\sqrt{3} \\mathbf{\\imath}\\right) \\sqrt[3]{\\frac{1}{27} a^{3} - \\frac{1}{6} a b + \\frac{1}{2} c + \\sqrt{\\left(- \\frac{1}{9} a^{2} + \\frac{1}{3} b\\right)^{3} + \\frac{1}{4} \\left(\\frac{2}{27} a^{3} - \\frac{1}{3} a b + c\\right)^{2}}}} - \\left(- \\frac{1}{2} + \\frac{1}{2} \\sqrt{3} \\mathbf{\\imath}\\right) \\sqrt[3]{\\frac{1}{27} a^{3} - \\frac{1}{6} a b + \\frac{1}{2} c + \\sqrt{\\left(- \\frac{1}{9} a^{2} + \\frac{1}{3} b\\right)^{3} + \\frac{1}{4} \\left(\\frac{2}{27} a^{3} - \\frac{1}{3} a b + c\\right)^{2}}}\\end{bmatrix}$$"
       ],
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "\u23a1                                                                             \n",
        "\u23a2                                                                             \n",
        "\u23a2                                                                             \n",
        "\u23a2                                       2                                     \n",
        "\u23a2                                      a    b                                 \n",
        "\u23a2                                    - \u2500\u2500 + \u2500                                 \n",
        "\u23a2  a                                   9    3                                 \n",
        "\u23a2- \u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - 3\n",
        "\u23a2  3            __________________________________________________________   \u2572\n",
        "\u23a2              \u2571                        _________________________________     \n",
        "\u23a2             \u2571                        \u2571                               2      \n",
        "\u23a2            \u2571                        \u2571                \u239b   3          \u239e       \n",
        "\u23a2           \u2571                        \u2571             3   \u239c2\u22c5a    a\u22c5b    \u239f       \n",
        "\u23a2          \u2571      3                 \u2571    \u239b   2    \u239e    \u239c\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500 + c\u239f       \n",
        "\u23a2         \u2571      a    a\u22c5b   c      \u2571     \u239c  a    b\u239f    \u239d 27     3     \u23a0       \n",
        "\u23a2      3 \u2571       \u2500\u2500 - \u2500\u2500\u2500 + \u2500 +   \u2571      \u239c- \u2500\u2500 + \u2500\u239f  + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500      \n",
        "\u23a3      \u2572\u2571        27    6    2   \u2572\u2571       \u239d  9    3\u23a0            4              \n",
        "\n",
        "        __________________________________________________________            \n",
        "       \u2571                        _________________________________             \n",
        "      \u2571                        \u2571                               2              \n",
        "     \u2571                        \u2571                \u239b   3          \u239e               \n",
        "    \u2571                        \u2571             3   \u239c2\u22c5a    a\u22c5b    \u239f               \n",
        "   \u2571      3                 \u2571    \u239b   2    \u239e    \u239c\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500 + c\u239f               \n",
        "  \u2571      a    a\u22c5b   c      \u2571     \u239c  a    b\u239f    \u239d 27     3     \u23a0       a       \n",
        " \u2571       \u2500\u2500 - \u2500\u2500\u2500 + \u2500 +   \u2571      \u239c- \u2500\u2500 + \u2500\u239f  + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  , - \u2500 + \u2500\u2500\u2500\u2500\n",
        "\u2571        27    6    2   \u2572\u2571       \u239d  9    3\u23a0            4              3       \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                          \u239b   \n",
        "                                                                          \u239c  1\n",
        "                                                                          \u239c- \u2500\n",
        "                                                                          \u239d  2\n",
        "\n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                     2                                        \n",
        "                                    a    b                                    \n",
        "                                  - \u2500\u2500 + \u2500                                    \n",
        "                                    9    3                                    \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                     _________________________________________________________\n",
        "                    \u2571                        _________________________________\n",
        "                   \u2571                        \u2571                               2 \n",
        "                  \u2571                        \u2571                \u239b   3          \u239e  \n",
        "                 \u2571                        \u2571             3   \u239c2\u22c5a    a\u22c5b    \u239f  \n",
        "     ___  \u239e     \u2571      3                 \u2571    \u239b   2    \u239e    \u239c\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500 + c\u239f  \n",
        "   \u2572\u2571 3 \u22c5\u2148\u239f    \u2571      a    a\u22c5b   c      \u2571     \u239c  a    b\u239f    \u239d 27     3     \u23a0  \n",
        " - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c53 \u2571       \u2500\u2500 - \u2500\u2500\u2500 + \u2500 +   \u2571      \u239c- \u2500\u2500 + \u2500\u239f  + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \n",
        "      2   \u23a0 \u2572\u2571        27    6    2   \u2572\u2571       \u239d  9    3\u23a0            4         \n",
        "\n",
        "                             _________________________________________________\n",
        "                            \u2571                        _________________________\n",
        "                           \u2571                        \u2571                         \n",
        "                          \u2571                        \u2571                \u239b   3     \n",
        "                         \u2571                        \u2571             3   \u239c2\u22c5a    a\u22c5\n",
        "    \u239b        ___  \u239e     \u2571      3                 \u2571    \u239b   2    \u239e    \u239c\u2500\u2500\u2500\u2500 - \u2500\u2500\n",
        "    \u239c  1   \u2572\u2571 3 \u22c5\u2148\u239f    \u2571      a    a\u22c5b   c      \u2571     \u239c  a    b\u239f    \u239d 27     3\n",
        "\u2500 - \u239c- \u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c53 \u2571       \u2500\u2500 - \u2500\u2500\u2500 + \u2500 +   \u2571      \u239c- \u2500\u2500 + \u2500\u239f  + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "_   \u239d  2      2   \u23a0 \u2572\u2571        27    6    2   \u2572\u2571       \u239d  9    3\u23a0            4 \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "\n",
        "_________                                                                     \n",
        "________                                                                      \n",
        "      2                                                                       \n",
        "     \u239e                                                    2                   \n",
        "b    \u239f                                                   a    b               \n",
        "\u2500 + c\u239f                                                 - \u2500\u2500 + \u2500               \n",
        "     \u23a0       a                                           9    3               \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500  , - \u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "             3                            ____________________________________\n",
        "                                         \u2571                        ____________\n",
        "                                        \u2571                        \u2571            \n",
        "                                       \u2571                        \u2571             \n",
        "                                      \u2571                        \u2571             3\n",
        "                 \u239b        ___  \u239e     \u2571      3                 \u2571    \u239b   2    \u239e \n",
        "                 \u239c  1   \u2572\u2571 3 \u22c5\u2148\u239f    \u2571      a    a\u22c5b   c      \u2571     \u239c  a    b\u239f \n",
        "                 \u239c- \u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c53 \u2571       \u2500\u2500 - \u2500\u2500\u2500 + \u2500 +   \u2571      \u239c- \u2500\u2500 + \u2500\u239f \n",
        "                 \u239d  2      2   \u23a0 \u2572\u2571        27    6    2   \u2572\u2571       \u239d  9    3\u23a0 \n",
        "\n",
        "                                                  ____________________________\n",
        "                                                 \u2571                        ____\n",
        "                                                \u2571                        \u2571    \n",
        "                                               \u2571                        \u2571     \n",
        "                                              \u2571                        \u2571      \n",
        "                         \u239b        ___  \u239e     \u2571      3                 \u2571    \u239b  \n",
        "                         \u239c  1   \u2572\u2571 3 \u22c5\u2148\u239f    \u2571      a    a\u22c5b   c      \u2571     \u239c  \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u239c- \u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c53 \u2571       \u2500\u2500 - \u2500\u2500\u2500 + \u2500 +   \u2571      \u239c- \n",
        "______________________   \u239d  2      2   \u23a0 \u2572\u2571        27    6    2   \u2572\u2571       \u239d  \n",
        "_____________________                                                         \n",
        "                   2                                                          \n",
        "   \u239b   3          \u239e                                                           \n",
        "   \u239c2\u22c5a    a\u22c5b    \u239f                                                           \n",
        "   \u239c\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500 + c\u239f                                                           \n",
        "   \u239d 27     3     \u23a0                                                           \n",
        " + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                                                          \n",
        "           4                                                                  \n",
        "\n",
        "______________________________\u23a4\n",
        "_____________________________ \u23a5\n",
        "                           2  \u23a5\n",
        "           \u239b   3          \u239e   \u23a5\n",
        "       3   \u239c2\u22c5a    a\u22c5b    \u239f   \u23a5\n",
        " 2    \u239e    \u239c\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500 + c\u239f   \u23a5\n",
        "a    b\u239f    \u239d 27     3     \u23a0   \u23a5\n",
        "\u2500\u2500 + \u2500\u239f  + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  \u23a5\n",
        "9    3\u23a0            4          \u23a5\n",
        "                              \u23a5\n",
        "                              \u23a5\n",
        "                              \u23a5\n",
        "                              \u23a5\n",
        "                              \u23a5\n",
        "                              \u23a5\n",
        "                              \u23a5\n",
        "                              \u23a6"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Differential Equations"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "A population that grows without bound is modeled by the differential equation\n",
      "\n",
      "$$f'(t)=af(t)$$\n",
      "\n",
      "Solve this differential equation using SymPy."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "dsolve(f(t).diff(t) - a*f(t), f(t))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{f}{\\left (t \\right )} = C_{1} e^{a t}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFwAAAAbCAYAAADxsuiMAAAABHNCSVQICAgIfAhkiAAAA/VJREFU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\nQmCC\n",
       "prompt_number": 7,
       "text": [
        "           a\u22c5t\n",
        "f(t) = C\u2081\u22c5\u212f   "
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If the population growth is bounded, it is modeled by \n",
      "\n",
      "$$f'(t) = f(t)(1 - f(t))$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "dsolve(f(t).diff(t) - f(t)*(1 - f(t)), f(t))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\log{\\left (\\operatorname{f}{\\left (t \\right )} -1 \\right )} - \\log{\\left (\\operatorname{f}{\\left (t \\right )} \\right )} = C_{1} - t$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAO4AAAAYCAYAAAD9JcEmAAAABHNCSVQICAgIfAhkiAAABrpJREFU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\n09PT07Nn8T9EbAbRBz88IwAAAABJRU5ErkJggg==\n",
       "prompt_number": 8,
       "text": [
        "log(f(t) - 1) - log(f(t)) = C\u2081 - t"
       ]
      }
     ],
     "prompt_number": 8
    }
   ],
   "metadata": {}
  }
 ]
}