{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solvers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For each exercise, fill in the function according to its docstring. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "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)"
   ]
  },
  {
   "cell_type": "markdown",
   "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",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def quadratic():\n",
    "    return solve(a*x**2 + b*x + c, x)\n",
    "quadratic()"
   ]
  },
  {
   "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",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def cubic():\n",
    "    return solve(x**3 + a*x**2 + b*x + c, x)\n",
    "cubic()"
   ]
  },
  {
   "cell_type": "markdown",
   "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",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dsolve(f(t).diff(t) - a*f(t), f(t))"
   ]
  },
  {
   "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",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dsolve(f(t).diff(t) - f(t)*(1 - f(t)), f(t))"
   ]
  }
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