{ "metadata": { "name": "" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Solvers" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from sympy import *\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. " ] }, { "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": [ "$$\\begin{bmatrix}\\frac{- b + \\sqrt{- 4 a c + b^{2}}}{2 a}, & - \\frac{b + \\sqrt{- 4 a c + b^{2}}}{2 a}\\end{bmatrix}$$" ], "metadata": {}, "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": [ "$$\\begin{bmatrix}- \\frac{1}{3} a + \\frac{- \\frac{1}{9} a^{2} + \\frac{1}{3} b}{\\sqrt[3]{\\frac{1}{27} a^{3} - \\frac{1}{6} a b + \\frac{1}{2} c + \\sqrt{\\left(- \\frac{1}{9} a^{2} + \\frac{1}{3} b\\right)^{3} + \\frac{1}{4} \\left(\\frac{2}{27} a^{3} - \\frac{1}{3} a b + c\\right)^{2}}}} - \\sqrt[3]{\\frac{1}{27} a^{3} - \\frac{1}{6} a b + \\frac{1}{2} c + \\sqrt{\\left(- \\frac{1}{9} a^{2} + \\frac{1}{3} b\\right)^{3} + \\frac{1}{4} \\left(\\frac{2}{27} a^{3} - \\frac{1}{3} a b + c\\right)^{2}}}, & - \\frac{1}{3} a + \\frac{- \\frac{1}{9} a^{2} + \\frac{1}{3} b}{\\left(- \\frac{1}{2} - \\frac{1}{2} \\sqrt{3} \\mathbf{\\imath}\\right) \\sqrt[3]{\\frac{1}{27} a^{3} - \\frac{1}{6} a b + \\frac{1}{2} c + \\sqrt{\\left(- \\frac{1}{9} a^{2} + \\frac{1}{3} b\\right)^{3} + \\frac{1}{4} \\left(\\frac{2}{27} a^{3} - \\frac{1}{3} a b + c\\right)^{2}}}} - \\left(- \\frac{1}{2} - \\frac{1}{2} \\sqrt{3} \\mathbf{\\imath}\\right) \\sqrt[3]{\\frac{1}{27} a^{3} - \\frac{1}{6} a b + \\frac{1}{2} c + \\sqrt{\\left(- \\frac{1}{9} a^{2} + \\frac{1}{3} b\\right)^{3} + \\frac{1}{4} \\left(\\frac{2}{27} a^{3} - \\frac{1}{3} a b + c\\right)^{2}}}, & - \\frac{1}{3} a + \\frac{- \\frac{1}{9} a^{2} + \\frac{1}{3} b}{\\left(- \\frac{1}{2} + \\frac{1}{2} \\sqrt{3} \\mathbf{\\imath}\\right) \\sqrt[3]{\\frac{1}{27} a^{3} - \\frac{1}{6} a b + \\frac{1}{2} c + \\sqrt{\\left(- \\frac{1}{9} a^{2} + \\frac{1}{3} b\\right)^{3} + \\frac{1}{4} \\left(\\frac{2}{27} a^{3} - \\frac{1}{3} a b + c\\right)^{2}}}} - \\left(- \\frac{1}{2} + \\frac{1}{2} \\sqrt{3} \\mathbf{\\imath}\\right) \\sqrt[3]{\\frac{1}{27} a^{3} - \\frac{1}{6} a b + \\frac{1}{2} c + \\sqrt{\\left(- \\frac{1}{9} a^{2} + \\frac{1}{3} b\\right)^{3} + \\frac{1}{4} \\left(\\frac{2}{27} a^{3} - \\frac{1}{3} a b + c\\right)^{2}}}\\end{bmatrix}$$" ], "metadata": {}, "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": [ "$$\\operatorname{f}{\\left (t \\right )} = C_{1} e^{a t}$$" ], "metadata": {}, "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", "$$f'(t) = f(t)(1 - f(t))$$\n", "\n", "Solve this differential equation using SymPy." ] }, { "cell_type": "code", "collapsed": false, "input": [ "dsolve(f(t).diff(t) - f(t)*(1 - f(t)), f(t))" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\log{\\left (\\operatorname{f}{\\left (t \\right )} -1 \\right )} - \\log{\\left (\\operatorname{f}{\\left (t \\right )} \\right )} = C_{1} - t$$" ], "metadata": {}, "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": {} } ] }