{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "# remember to include these two lines of code at the start of your document!\n",
    "from IPython.core.interactiveshell import InteractiveShell\n",
    "InteractiveShell.ast_node_interactivity = \"all\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# MATH1110 Lab 5: Differential Equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 11 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x, y = var('x, y')\n",
    "backdrop = plot(10, x, 0, 4*pi, color = 'lightgrey', fill = True, fillcolor = 'lightgrey', fillalpha = 0.2)\n",
    "p1 = plot(cos(x)^2+sin(2*x)+1, x, 0, 4*pi, fill = True, fillcolor = 'skyblue', color = 'dodgerblue')\n",
    "p2 = plot(cos(21*x)/12+2, x, 0, 4*pi, fill = True, fillcolor = 'lightblue', color = 'aqua')\n",
    "p3 = plot(1.3*e^sin(2*x+3)/2, 0, 4*pi, fill = True, fillcolor = 'steelblue', color = 'navy')\n",
    "p4 = plot(e^sin(x+1)/2+0.4, 0, 4*pi, fill = True, fillcolor = 'dodgerblue', color = 'slateblue')\n",
    "pts = points( [[1, 4.2], [3.5, 6], [4.8, 5.9], [6.4, 6.1], [7.8, 4.5], [10.4, 8.5], [10.8, 6.2]], \\\n",
    "             color = 'yellow', marker = '*', size = 90)\n",
    "show(backdrop+p1+p2+p3+p4+pts, ymax = 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Objectives for today:**\n",
    "\n",
    "1. Derivatives\n",
    "2. Differential equations\n",
    "3. Initial value problems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Derivatives\n",
    "\n",
    "As you've been learning in lectures, completing derivatives by hand can be quite tricky. In Sage, it is mercifully easy! The command is *derivative()* or *diff()* (reminder that all the commands for the course are described in the *Glossary of Commands* on blackboard!), which takes two arguments: (1) the function you're differentiating, and (2) the variable you're differentiating with respect to."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{4^{x - \\log\\left(x\\right)} \\arctan\\left(x\\right)}{\\sin\\left(x + e^{x}\\right)}</script></html>"
      ],
      "text/plain": [
       "4^(x - log(x))*arctan(x)/sin(x + e^x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{2 \\cdot 4^{x - \\log\\left(x\\right)} {\\left(\\frac{1}{x} - 1\\right)} \\arctan\\left(x\\right) \\log\\left(2\\right)}{\\sin\\left(x + e^{x}\\right)} - \\frac{4^{x - \\log\\left(x\\right)} {\\left(e^{x} + 1\\right)} \\arctan\\left(x\\right) \\cos\\left(x + e^{x}\\right)}{\\sin\\left(x + e^{x}\\right)^{2}} + \\frac{4^{x - \\log\\left(x\\right)}}{{\\left(x^{2} + 1\\right)} \\sin\\left(x + e^{x}\\right)}</script></html>"
      ],
      "text/plain": [
       "-2*4^(x - log(x))*(1/x - 1)*arctan(x)*log(2)/sin(x + e^x) - 4^(x - log(x))*(e^x + 1)*arctan(x)*cos(x + e^x)/sin(x + e^x)^2 + 4^(x - log(x))/((x^2 + 1)*sin(x + e^x))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y = 4^(x-ln(x))*arctan(x)/sin(x+e^x)\n",
    "show(y)\n",
    "show(diff(y, x)) # I would not want to try doing that by hand!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is an optional third argument in which you can indicate the number of times you're taking the derivative (the default is 1, of course). For example, if you needed the third derivative $\\large{ \\frac{d^3y}{dx^3} }$, you would put the number 3 in the third argument.\n",
    "\n",
    "Find the first and second derivatives of $f(x) = \\large{ \\frac{\\sin{(x)}}{4x} }$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Any questions? Hopefully this part was pretty straight-forward."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Differential Equations\n",
    "\n",
    "In differential equations; variables, functions, and the derivatives of these fucntions can be intertwined into an equation together. Linear, ordinary differential equations that can be solved with the command **desolve**.\n",
    "\n",
    " We previously learned about the **solve** function, and **desolve** works in a very similar way. The first entry is for our equation, but the second entry is the dependent variable (often $x$, or whatever takes its place in the $\\frac{dy}{dx}$ notation). Our goal is to figure out what the function $y$ is.\n",
    " \n",
    "The first step is to set up your variables so that one is assigned as a **function** of the other. It looks like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}f\\left(x\\right)</script></html>"
      ],
      "text/plain": [
       "f(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# for some fuction f using variable x, we would write:\n",
    "x = var('x')\n",
    "f = function('f')(x)\n",
    "show(f)\n",
    "# now f is an arbitrary function of x."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It will appear this way throughout your outputs, so don't be confused and think it's multiplying the function $f$ by a variable $x$. It is just indicating that $f$ is a *function of* $x$.\n",
    "\n",
    "A differential equation could look something like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\partial}{\\partial x}f\\left(x\\right)</script></html>"
      ],
      "text/plain": [
       "diff(f(x), x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(x + 3\\right)} \\frac{\\partial}{\\partial x}f\\left(x\\right) = e^{f\\left(x\\right)}</script></html>"
      ],
      "text/plain": [
       "(x + 3)*diff(f(x), x) == e^f(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "df = diff(f, x)\n",
    "show(df) # this is the derivative of an arbitrary function f with respect to x.\n",
    "# Sage's presentation looks a little weird, but you can imagine the f(x)\n",
    "# being in the numerator of the fraction instead of on the side.\n",
    "\n",
    "show( (x+3)*df == e^f)\n",
    "# don't forget the double == !"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice how the derivative of $f$, $\\frac{df}{dx}$, is part of an equation that involves $f$ and $x$.\n",
    "\n",
    "\n",
    "Let's now set up the differential equation $\\large{ \\frac{dy}{dx}=y }$. In other words, what function $y$ is equal to its own derivative?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The $\\_C$ part just means that you could put a coefficient in front and the equation would still hold.\n",
    "This is what we call a *general solution* to the differential equation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One more example. What is the general solution of $f$ for which $\\large{ \\frac{df}{dx} } = \\frac{1}{3}f^{-2}$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this case, Sage leaves us with a tiny bit of work to do. We can isolate $f(x)$ by taking the cube root on both sides.\n",
    "\n",
    "Just watch out for instances where you function isn't completely isolated in your output!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise questions\n",
    "\n",
    "Using the approach from above, solve for the general solutions for the functions in the given differential equations.\n",
    "\n",
    "1. $\\large{ \\frac{dy}{dx} = x^2+x }$\n",
    "\n",
    "\n",
    "2. $2 \\large{ (\\frac{dh}{dx}) = h+x }$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Initial Value Problems\n",
    "\n",
    "In the previous exercises we always get a constant $C$ in our output. $C$ can be any real number, which means there are an infinite number of versions of the function which satisfy the differential equation. Initial value problems require us to find a unique solution from that general group of solutions which satisfy initial conditions (\"ics\") that are provided in the question.\n",
    "\n",
    "We will again use **desolve** with the added argument **ics = [a, b]** to get the specific solution. Here **a** is the initial condition of the independent variable, and **b** is the initial condition of the funciton. So if we had a differential equation using $\\frac{dy}{dx}$, **a** would refer to $x$ and **b** would refer to $y$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using the same $\\large{ \\frac{df}{dx} }= \\frac{1}{3}f^{-2}$ as before, now solve for a unique solution where $x=1$ and $f=-1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# copy and paste your work from above\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "scrolled": true
   },
   "source": [
    "What if $x=2$ and $f = 4$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In summary, here are the steps to solving a differential equation in Sage:\n",
    "1. Define your independent variable with *var(\\_)*, and the function of that variable with *function(\\_)*.\n",
    "2. Define the derivative term (i.e. $\\frac{dy}{dx}$ or $\\frac{dg}{dx}$, whatever it may be).\n",
    "3. Recreate the differential equation given in the problem.\n",
    "4. Put the equation in the first argument of *desolve(\\_)*, and the function in the second argument.\n",
    "5. Identify initial conditions in the third argument, if asked for a particular solution.\n",
    "6. Check your output to ensure your function has been completely solved for."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise questions\n",
    "\n",
    "3. Using the differential equation from Exercise question 1, find the unique solution that accompanies initial conditions $x=-2$ and $y=1$.\n",
    "4. Using $2 \\large{ (\\frac{dh}{dx}) = h+x }$ from Exercise question 2, find the unique solution for $h$ where $x=0$ and $h=0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is **NO** lab next week. We'll see you here again after reading week :)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Reminders\n",
    "\n",
    "> Regular office hours generally are cancelled over reading week. If you'd like in-person (or zoom) support, email ahead with your prof or TA to confirm their availability.\n",
    "\n",
    "> Maya will be out of town next week (14 - 17 october), but will be checking email and holding a virtual office hour on wednesday the 15th if you need support. More details will be in a Blackboard announcement ... eventually.\n",
    "\n",
    "> Assignment 3 is due 17 october."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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