Lecture02.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lecture 2: second-order ordinary differential equations\n",
    "\n",
    "We now look at solving second-order ordinary differential equations using a computer algebra system."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To use SymPy, we first need to import it and call `init_printing()` to get nicely typeset equations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "\n",
    "# This initialises pretty printing\n",
    "init_printing()\n",
    "from IPython.display import display\n",
    "\n",
    "# This command makes plots appear inside the browser window\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mass-spring-damper system\n",
    "\n",
    "The differential equation that governs an unforced, single degree-of-freedom mass-spring-damper system is\n",
    "\n",
    "$$\n",
    "m \\frac{d^{2}y}{dt^{2}} + \\lambda \\frac{dy}{dt} + ky = 0\n",
    "$$\n",
    "\n",
    "To solve this problem using SymPy, we first define the symbols $t$ (time), $m$ (mass), $\\lambda$ (damper coefficient) and $k$ (spring stiffness), and the function $y$ (displacement): "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "t, m, lmbda, k = symbols(\"t m lambda k\")\n",
    "y = Function(\"y\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that we mis-spell $\\lambda$ as `lmbda` because `lambda` is a protected keyword in Python.\n",
    "\n",
    "Next, we define the differential equation, and print it to the screen:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARYAAAAmCAYAAAABQLRWAAAABHNCSVQICAgIfAhkiAAACOVJREFU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\nCQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$k y{\\left (t \\right )} + \\lambda \\frac{d}{d t} y{\\left (t \\right )} + m \\frac{d^{2}}{d t^{2}}  y{\\left (t \\right )} = 0$$"
      ],
      "text/plain": [
       "                          2          \n",
       "           d             d           \n",
       "k⋅y(t) + λ⋅──(y(t)) + m⋅───(y(t)) = 0\n",
       "           dt             2          \n",
       "                        dt           "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eqn = Eq(m*Derivative(y(t), t, t) + lmbda*Derivative(y(t), t) + k*y(t), 0)\n",
    "display(eqn)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Checking the order of the ODE:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "This order of the ODE is: 2\n"
     ]
    }
   ],
   "source": [
    "print(\"This order of the ODE is: {}\".format(ode_order(eqn, y(t))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and now classifying the ODE:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Properties of the ODE are: ('nth_linear_constant_coeff_homogeneous', '2nd_power_series_ordinary')\n"
     ]
    }
   ],
   "source": [
    "print(\"Properties of the ODE are: {}\".format(classify_ode(eqn)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "we see as expected that the equation is linear, constant coefficient, homogeneous and second order."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `dsolve` function solves the differential equation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAfcAAAAjCAYAAACJrUOqAAAABHNCSVQICAgIfAhkiAAAC6lJREFU\neJztnXuQHEUdxz9HCOTMYZRASXgZFRNCAUZCHpgQY0BSIiJKoSVU5EJBChBKQLQMYIgCRUSLZyQB\ntbLxgRgSJOAzIoRHUAhgJA8EqeJAUXkIEp6a4PnHt6dudrfntTu7s3v7+1RN3V53b09vT3+ne379\n6x4wDMMwDGNQ0VV0AYxM7Ay8D1hXdEHqoL/oAgwCTLeti2nUANOokYEhwEJgu6ILUgd7AicWXYg2\n50Cgt+hCGF5MowaYRo2MnALMLLoQdTIXGFV0IQYBVwO7FF0IowrTqBFgGjVSMQy4l/Y39VxYdAEG\nCdOBK4ouhFGGadQIU7hG29l81EkcDmymvefChgFvFF2IFmcvYA261o8Ax0Wkuxf4FKbfVsI02hmY\nRo1cWQKcWnQh6uRIYGLRhWhxRgHj3efdgGeA4RFp7wM+1IxCGakwjXYGbaNRG1W0B5OAx4suRJ1M\nAB4suhAtzj+A9e7zP4EXkPe1j8eByc0olJEK02hn0DYatc69PXgXsCWHfBYA/wVuoflzg100x2S5\nEPhtE85TCwtIX/8TkPf1XyPit6B2YbQGptH0mEaNjuNbwG884f8BxuaQ/9uBk5GAxyekzZMD0PxT\nXsxDv2GRJ+7XqB5bkbT1vzOwiXiT3sXA9yPilgBXonpah24yzwO3AftnK7LhwadT02g1UTo1jTZB\no/bk3lpMAh7whL+GnF3qZQtQQjeiA3LIz8d7gEsqwj5KfiP1KWi5ziMR8eOBP+Z0rjSU0Gg/DWnq\nf0f01LAQzdlFMQx4xRPeBRzt8pgBXItuQDOBbcDtRJsRjXT4dFq0Rkukb4eN1ijE69Q02gSNWufe\nGuyATEHTgQvQqHFzKP5ZNKLMg6HAqzTuCe4AtIFDTyisx52zXkYAPwZOAl7yxO+GzGDBnNhw4Ebg\nYWA02qCjH/gscAfwOvAnYF/gYOBuF/YAsHcO5fURV/9d6MZyB/DDhHxGoHZRyUR087kXmAUsBTYC\nG4DZwK7A1FD6VqiTdiFOp6bRAeJ0mqRRKL5NDgqN5tW5LwOeI9prsJIJrrAn53T+dmcbcIj7PBl5\nZIYv7hPkt7HEfGAkjXtyvxX4OXCE+38kcjrJg+uBFcCdEfHj0VKex5CJ9AFUt1OBPuADLt1pwNdR\nXe+IxHoZcB66DrsA5+RU5kri6n8qEvAx6Oa3PiId6Cbpc+A6BvgF+t2V7IQ0H77htkKdtAtxOjWN\nDhCn0ySNQvFt0jTqmAj8LybTs1FHfnxF+M+Q52FP1TeK4f3ApcBDaO5jq/t7O3A68LYGn/8oZBLy\nOXGc7spWL/sDbyJxP51DflEchUajACcA780hz1PQtRnq/l9D9VzeV4H7gWOBfwFnVcSfD/ybcieX\na9B1HhkKWwr8NGW5SqQ3+eVZ/0/if1LcBHw64jvLkTl0SCisEXXSKIrWKETrtGiNlkjfDqExGoVk\nnSZpFPJvkyVMozWx2p24OyL+R6hzr3Q2meTCz8uzMDXQBVyEzG39wFpgMZpP+QlqgP3APQ0ux9di\nzrEnuoFVcjEqW9wxw6XtQvND16D5nn5kNvKRJV8f3ahhb+d+VxRpzzMWNeRwG1pDded+I/Ai8DLw\nYc/5lqP2GGYVcF1F2O/Q9fdxHjLZBcdW1HbCYYd6vpel/pPYG/iVJ3wf9FTks6BdDvyd6pt4HnXS\naFpFoxCt02ZrtNZ2GJC3RiGdTpM0CvW3SdNoDoxBT+3Xx6R5FDkV+J5IHwWeoti5/6XoIm5iwPwR\nphuNnJZ64vJkJdqPOIrFVM8v7YLmXeKO4GnmNLRkYyc0t9VPuem/1nyjuA2YhrxBo0h7nl5X3m2h\nox+1vW3IRAXwZ+B7aP7J5/n7OHBGRdjTVL/k4QVkevOxMxJocATXLRzmG+hmqf8kzmfAPBzmXGRu\nreQKZCXb1xOXR53E0UvyYDCJVtEoxOu0mRqttR2GyVOjkE6nSRqF+tukabSCc9CP+VJEYcciD8K7\nQ2EL3XcO86QP4nzHbJfmQvf/rIhzNprgN29GFzSOJKFEcSwawb2ARo9/QSPLIRXpniDeB2EUmguq\nhVHIunJ0KOxlGruj1lz0hBM1Os/CO5C5LHysA25wn7vQDeYtNE10AhqdHxTKY7iLD4t1JLr+4Q5j\nLxc2JmXZSiSb/PKs/+HAtyPi7qG6DV2FNtsYF5FXI+okTC/1de6tpFGI12mRGi2RzSwP+WoUknUa\ntLcojUJj2mSJDtfoZBd5U0ShVqPR14GhsAddmM/E8BlUqYEZbUHo2MulOdzFR1VEIxmFzCNbaYxX\n6hBkMuxHN4vFaF3jYy5sWUX6PlQPuyOR+JhBef2n5Saqr+ta/OvE82J3ZC7dvkH5r6G8/FNQWwxu\n8BehrSH3cP8fgkQSbquHofm1cBk/gSxNaa1JJZJvHHnW/xz8ndyuqC2H5+W+g+aHZyLnnuAI/Fwa\nVSdheqm9c281jUKyTmdQjEZLZO/cG61RKNdpkkahMW2yRIdrdCgylficCI5Djf2qUNhwdKE2+DJz\nzHXfmxsRP8LF+9Z2V3IW5QOEpOOYhPy+6c7dKFPeIpf/pZRfhKGo0fQD+4XCTwD+hkxYi2Pyzfp0\n8nE0Iq305F0M3JUxr6zMTk5SM2soF96paJonoAvNVT2EnupPRSbBMOdSvd3mBWiJSlpKxN848q7/\nqOt/EtVrbqMsZwtcfKPqJEwvtXfuraZRSKfTIjRaInvnDo3VKJTrNEmjQZq822QJ0yh3uYzDP3I4\nmod4lnLngjEu7eqY/Ja4NAfHpHkDmSSS6CPZsSN8lBLyC0bnH0lx7qxMRuK/JSI+GPTMacC5jc5k\nFfCVogvhoZfaO3fTqDGYaKpGK80xa9EGDYcAN7uw+cgTdA6agwgIXPB9m4kEHITMEHFP9y+Sbv/d\n0SnSpKWHgcHJ/Rm/Ox2NnCYgs9YcqgcSZ6JR6ev4R4yBidE2ETLyYi0yMRdJH/DuiLg7PWHLqHYK\nCqhHo/PQUqPAT+gPLmxjKI1p1Gg2TdWor3MHjWpvRp57ZwO/p3r+KXjvb9SWi9ujxf2bkcCi6Kb5\n7xDe1f3dgsSdhR50k/iBO3wEm0N8LiGvpzKe2zCiuKzoAqD56sp56PHAJ9H9o68ibj3R1KPRGWhL\nz3WoA/8GWqa2H3qYANOo0XyaqtHKzv0+NFKe4v5fhJxOvuDCwzzn/o7Ez36o43845vzboZvBkynK\nehbRjmY+1hNtcnvN/e1Gv++tDPn+0h3gN/0PQzemu8nPC9Uw2oErPWG9qHMvofnXtNSj0crVN7OR\n1XEqWv5lGjU6kk1oicLxqEO/NiJdF+rgn4+I/7z7/pkx5xrn0qxMUa4+8p1zD/LzLeMLE2eWe5Vq\ns2I3A2tyDaPT6aX2Ofc+6tcoyIeoH63pBtOo0aFchxr+K6jjfmdM2hUu7T6euGDb2RNjvj/Hpalc\noN8MvujO/Rj+jQO6gI8hb84ofJ07aFP/fqK3GJyGfw2tYQw2eqm9c89Do+Df0tM0agxqfOsb1yJP\n0R7UQcc5zK1EG0DMQps7hHnI/b0EOae8hkbK4bWERyBz26qsBc+Bq9F61JOQw99qtBPQVrQOfxpy\nJLyhhry/jF4MsBLN9T2Cni72QI54Q7E3bBlGEnlo9HKXbhrlpn3TqNFxHMrA2nPflrFhdkBL5KK8\nWc9Ao+43XZ7hdwiPQI50UfPizeJI9BKbZ9DuVC+iQchyZHWI8imA6Cd30C5MK9Ayv61o96sNyDKS\nZGY0jMFCL/VvP1urRuO29ATTqNFh3MrAFoFpmIfE+8GM5zmT8nmwdiSuczcMozjitvQ0jI4jcKLL\nsuXeMLRc5LYM3+lGb79ZkeE7rUIPWt4zHi3Rme8+mwnPMFqDpC09DaMj2Bu9Y/e7aD36RrK/F3k6\negmMb495H+PQxhGjM56nFZhBbd75hmE0h6QtPQ2jIwi2WXwJzWHtXmxxDMMwDMMwDMMwDMMwDMMw\nDMMwDMMwDMMwDMMwDKP5/B+bcEyLJLht+gAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$y{\\left (t \\right )} = C_{1} e^{\\frac{t}{2 m} \\left(- \\lambda - \\sqrt{- 4 k m + \\lambda^{2}}\\right)} + C_{2} e^{\\frac{t}{2 m} \\left(- \\lambda + \\sqrt{- 4 k m + \\lambda^{2}}\\right)}$$"
      ],
      "text/plain": [
       "             ⎛        _____________⎞         ⎛        _____________⎞\n",
       "             ⎜       ╱           2 ⎟         ⎜       ╱           2 ⎟\n",
       "           t⋅⎝-λ - ╲╱  -4⋅k⋅m + λ  ⎠       t⋅⎝-λ + ╲╱  -4⋅k⋅m + λ  ⎠\n",
       "           ─────────────────────────       ─────────────────────────\n",
       "                      2⋅m                             2⋅m           \n",
       "y(t) = C₁⋅ℯ                          + C₂⋅ℯ                         "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y = dsolve(eqn, y(t))\n",
    "display(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The solution looks very complicated because we have not specified values for the constants $m$, $\\lambda$ and $k$. The nature of the solution depends heavily on the relative values of the coefficients, as we will see later. We have four constants because the most general case the solution is complex, with two complex constants having four real coefficients.\n",
    "\n",
    "Note that the solution is make up of expoential functions and sinusoidal functions. This is typical of second-order ODEs."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Second order, constant coefficient equation\n",
    "\n",
    "We'll now solve \n",
    "\n",
    "$$\n",
    "\\frac{d^{2}y}{dx^{2}} + 2 \\frac{dy}{dx} - 3 y = 0\n",
    "$$\n",
    "\n",
    "The solution for this problem will appear simpler because we have concrete values for the coefficients.\n",
    "\n",
    "Entering the differential equation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASYAAAAmCAYAAABpszZuAAAABHNCSVQICAgIfAhkiAAACBhJREFU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\niqIoiqIoiqIoiqIoShXzPyQ2w6arp5iUAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- 3 y{\\left (x \\right )} + 2 \\frac{d}{d x} y{\\left (x \\right )} + \\frac{d^{2}}{d x^{2}}  y{\\left (x \\right )} = 0$$"
      ],
      "text/plain": [
       "                         2          \n",
       "            d           d           \n",
       "-3⋅y(x) + 2⋅──(y(x)) + ───(y(x)) = 0\n",
       "            dx           2          \n",
       "                       dx           "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y = Function(\"y\")\n",
    "x = symbols(\"x\")\n",
    "eqn = Eq(Derivative(y(x), x, x) + 2*Derivative(y(x), x) - 3*y(x), 0)\n",
    "display(eqn)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solving this equation,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMsAAAAaCAYAAAAZmai5AAAABHNCSVQICAgIfAhkiAAABlVJREFU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\nS2B2EW1KSEhISEhISEhISEgoGf4BWpX/TiTWRv4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$y{\\left (x \\right )} = C_{1} e^{- 3 x} + C_{2} e^{x}$$"
      ],
      "text/plain": [
       "           -3⋅x       x\n",
       "y(x) = C₁⋅ℯ     + C₂⋅ℯ "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y1 = dsolve(eqn)\n",
    "display(y1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "which is the general solution. As expected for a second-order equation, there are two constants."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the general solution is of the form\n",
    "\n",
    "$$\n",
    "y = C_{1} e^{\\lambda_{1} x} + C_{2} e^{\\lambda_{2} x}\n",
    "$$\n",
    "\n",
    "The constants $\\lambda_{1}$ and $\\lambda_{2}$ are roots of the \\emph{characteristic} equation\n",
    "\n",
    "$$\n",
    "\\lambda^{2} + 2\\lambda - 3 = 0\n",
    "$$\n",
    "\n",
    "This quadratic equation is trivial to solve, but for completeness we'll look at how to solve it using SymPy. We first define the quadratic equation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJUAAAAWCAYAAADJht45AAAABHNCSVQICAgIfAhkiAAABBNJREFU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\nckVFRUVFRUVFRXv8C6TIVFU9skapAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\lambda^{2} + 2 \\lambda - 3 = 0$$"
      ],
      "text/plain": [
       " 2              \n",
       "λ  + 2⋅λ - 3 = 0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eqn = Eq(lmbda**2 + 2*lmbda -3, 0)\n",
    "display(eqn)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and then compute the roots:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAAAVCAYAAADCUymGAAAABHNCSVQICAgIfAhkiAAAAg9JREFU\nWIXt2L1rFEEYx/GPYmGh+EIwQRAhkFQBRWwEwRM7ISBiaZHG/yB/wFkEo61iKSLa2CjY2HkBEUEw\nwUJFEc9CUVREBRWJxGI2ZKO7lxvJ7F7kvjDMvszs8+PZYZ5nHvpURgMLufa0VjVriwHLfbew+GJD\nweAZtPAhgZCz2I/RTNR3vMJNXMDHBDb/lRM4hL3Yg824hpMFY7/hdHY9gd1FH2wInm+uqszl/MR9\nXMI0zuNBZvc1diW0HcucoOsrnmTXV7uY15JbwXka0jt4Y8nzqcz2xYS2YzmMEayz5JtoB69PIKwT\nP0qeX8/6kaqEdMEdPFeyGrulageXMZ71j2pVkYCiIFcFk9iELULQOyg4d7omPcmo08GDufvbQvR9\nX4uahMRsEW1/53qdWqeAMCQEjyEcxzBmsS9K/RogZgW/UB6kinjTxZh3uIGHeIYrGIuw0fPEOPhI\nMhXhsPFYSOoHpDnk1EKvZBGwM+t/1apilanSwaNC1lCkYQo7cA+f/nh/WdjTJxJqS0aVWcRRnMFd\nvBTqDoPCeX8Yb3GqYN7iIpivQGOeY1kjBGM4IPxwwjY2GfPBhrRH5TGhoDMniJvHZ6EW0cT2knmz\n+IJtiXSV0dQ5S2qXzGupsRYRy1ZhTz5Xt5AIWlZwcC/Vg8eF1HBopYE101U9uG2ppklvpEq3lFfg\neol8PbhPn/+I3wrpgkm3jO7RAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ -3, \\quad 1\\right ]$$"
      ],
      "text/plain": [
       "[-3, 1]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(eqn)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "which as expected are the two exponents in the solution to the differential equattion."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}