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   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Solving Ordinary Differential Equations (ODEs) using Python"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Simple differential equations can be solved numerically using the Euler-Cromer method, but more complicated differential equations may require a\n",
      "more sophisticated method. The \"scipy\" library for Python contains numerous functions for scientific computing and data analysis. It includes the function\n",
      "\"odeint\" for numerically solving sets of first-order, ordinary differential equations (ODEs) using a sophisticated algorithm. Any set of differential equations can be written in the required form.  \n",
      "The example below calculates the solution to the second-order differential equation,\n",
      "$$\n",
      "\\frac{d^2x}{dt^2} = ax + b\\frac{dx}{dt}.\n",
      "$$\n",
      "It can be rewritten as the two first-order differential equations,\n",
      "$$\n",
      "\\frac{dx}{dt} = \\dot{x}\n",
      "$$\n",
      "and\n",
      "$$\n",
      "\\frac{d\\dot{x}}{dt} = ax + b\\dot{x}.\n",
      "$$\n",
      "Notice that the first of these equations is really just a definition. In Python, the function $x$ and its derivative $\\dot{x}$ will be elements of an array. The function $x$ will be the first element $x[0]$ (remember that the lowest index of an array is zero, not one) and the derivative $\\dot{x}$ will be the second element $y[1]$. The index indicates how many derivatives are taken of the function. In this notation, the differential equiations are\n",
      "$$\n",
      "\\frac{dx[0]}{dt} = x[1]\n",
      "$$\n",
      "and\n",
      "$$\n",
      "\\frac{dx[1]}{dt} = ax[0] + bx[1].\n",
      "$$\n",
      "The \"odeint\" function requires a function (called \"deriv\" in the example below) that will return the first derivative of each of element in the array. In other words, the first element returned is $dx[0]/dt$ and the second element is $dx[1]/dt$, which are both functions of $x[0]$ and $x[1]$. You must also provide initial values for $x[0]$ and $x[1]$,  which are placed in\n",
      "the array $yinit$ in the example below. Finally, the values of the times at which solutions are desired are provided in the array $time$. Note that \"odeint\" returns the values of both the function $x[0]=x$ and its derivative $x[1]=dx/dt$ at each time in an array $x$.  The function is plotted versus time."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy.integrate import odeint\n",
      "from pylab import *  # for plotting commands\n",
      "\n",
      "def deriv(x,t):  # return derivatives of the array y\n",
      "    a = -2.0\n",
      "    b = -0.1\n",
      "    return array([ x[1], a*x[0]+b*x[1]])\n",
      "\n",
      "time = linspace(0.0,100.0,1000)\n",
      "xinit = array([0.0005,0.2])  # initial values\n",
      "x = odeint(deriv,xinit,time)\n",
      "\n",
      "figure()\n",
      "plot(time,x[:,0])  # x[:,0] is the first column of x\n",
      "xlabel('t')\n",
      "ylabel('x')\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
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6WV9/Xc7dq69KOJjJyJDv329+I3/ZDxkib8gbNsjP0ldfNbwUt8HjkccnT5ZK\nZ+FCOVdt2sgx/PCH0p1odsnzTp0kFO64Q6rrV14BFi+W1x05EvjrX+Vz+H+tHo8c4/nnyx8MW7ZI\nt2xpqZyDu++Wtv5jqJdemoVLL83y+TzzzE+GDmVj+PDh6uGHH1a1tbWqrKxMTZ48WU2ZMsWuma1T\np06pvn37ql27dqmamho1YsQIVVhY2OA57777rpo0aZJSSqlPPvlEjR07VrutkhLH8vV79lSqosL8\nsR//WKnnnjN/7PRppTweperrzR+/4gql3n7b/LG9e5Xq0cPykFRqqlI7d5o/9tlnSmVkmD9WV6dU\nmzZybGY++ECpSy81f2zHDnldK6+8otS0aeaP/eMfSuXkWLddtEipe+4xf2zhQqUefNC67WOPKfWz\nn5k/NmWKUkuXWrd96CGl5s83f2zgQKU2b7Zu++CDSj3+eOP7q6qU6tzZ+hwrpdR998nX5W/5cqXG\nj7dup5RSc+Yo9T//0/j+u+9W6te/Dtz2N79R6vbbG95XU6NUUpJSGzcGbvv660qNGdPw5/mNN5Qa\nPFipU6cCt/34Y6USE5WqrJTb1dVKjRyp1O9+F7idUvJzl5ys1COPKLVpk3ydgwdb/076OnVKqZkz\nlYqLU+r88+V3+a9/tW9n+OADpW65RamrrlLqqaeUOnxYv211tVIffqhUXp5S+/bptzPU11u/d7Qk\nGtFg3dbuCQUFBY3uW7JkSdAv6GvlypVqwIABql+/furx736Tn3vuOfWczzv6Pffco/r166eGDx+u\nPv/884Bt/QU6McnJSn37rfljt92m1MsvWx93bKxSJ0+aP3bppfJDa6a6Wqn27a0/b48e3l9Qf19/\nrVT//uaPHT6sVMeO1p/388/ll89MQYFSo0ZZt333XaUmTDB/7A9/UOrOO63bvv66UtddZ/7Y7NlK\nPfOMddu//lWpG24wfywjQ6n//Me67R//KN9Df8ePK9WunVK1tYFf1+xvprw8pbKyrNspJW/GV17Z\n+P5Zs+SNP5C1a5UaOrThfSdPys+E1R8ahooKpbp2bfjm+NxzSk2cGLidUvIHyeDBSv3973J7/36l\neveWwNDxyCNKpafLz8KYMRJuum+apaVK3XijBP1ddyl14IBeO8OePUp9+qlSJ044a0f2mjRUWrNA\nJ6ZfP6W2bzd/7MYblfrb36w/b5cuSh08aP7Y2LFKrVtn/lhdnVQ5dXXmj7dvL8FjprxcqYQE88dK\nSuQvRitpq0EGAAAZPklEQVQ7d1pXI++9p1R2tnXbggL569NMbq5SDz9s3fbjj5W64ALzx665Rt6E\nraxdK29S/urrpWKoqgr8umZtP/tMqeHDrdsppdTWrUr16dP4/kceUernPw/ctrJS3tx9q5n6eqXO\nPVepLVsCtz19Wqn4eKWKirz3vf66dYXpb+pUpX71K/n4+HGlzjvP+ufQ37p1El4LF8ofH//v/+m1\nU0q+vmXLlJo+XamXXrL+2abWJZRQceWKesB+oN5qTAUIPK040JhKmzbWA/11dfI5rdoGGlMJNJ4C\nBB5TCTSdGAg8pdhuTKUpBur375c+6Lg467bGmIr/QHCgmV+GAQPk6/U/Xx9/3HjMwl98vAxi++4D\n9tVXMj5gDP5aOessWfX+wgve+154QQZZdTz+uMz0evNNaXPRRY2nIFsZN07GVb75RsZK7KYL+/J4\nZEHlK69I/38YhluplXPtj0CgUAk0pRgIPFgfaPYXYL0A8tgxeczql7JDBzkus+mFgWZ+AfJYdbX5\n7KJACx8Bb6iYzfLRCZWKCvO2dqHSq5eEpf+5CjTzy9C9u3x//bdc0QmVNm1kkNN3avGpUxIUOm/S\nV1/dcFHhihUyCK4z2fHuu4E//UkmN/zrX/K1Tpli3w6QaaWvvSazhc46y/l2/KNHS5s77tA7ViIr\nDBUTgaYUA4GnFQeqVADrBZBGqFjxeORxs2rFrlJp00Zmkxw+3PixQGtUAHlNj8f8mO1CpX17+ef/\nV/+JE3IsgdparVWx2p7FX3q6LAjzpRMqgMzeWrvWe/uLL6RyClQdGa6/Xt7c6+okxF95RaaN6ujb\nVyqF8eNlCrEx20fX+PEyE+rPfzZfGErUHBgqJkLp/gq2Ugm0RsVg1QVmV6kA1gsg7bq/AOsFkHah\nAki14j+teM8eWaBm11Vi1gVmtT2LvxEjJAwMSsltnVDx3yDy/fdlTYGOkSOlynrjDemK6tjRem2L\nmV/8QqaevvFGw2m+RK2Fa0Ml0KaSNTWB/0IMtP9XsJWKTqhYLYAMtEWLwWoBpF33F2C9AFI3VPzH\nVXbvDtz1ZTALFd1KxX8fr5ISWYvTq5d924svlm1VjBB++21Za6DriSdkweFdd8nqcyfdSR6PrC35\n3vf02xC1JK4NlaaoVOrr7cdj7MZUArGqVOy6vwDrwXq77i/AfLD+xAkJZbswS0w0DxX/PbDMmF2s\nq6hIL1RGjpSuIMO6dbJiWucNvmNHWfy4bJns7rtrl/XCRTPf+55sHvn++87aEUUD2xX10cpuoN6u\nUjELFWMsJlC3jtWqet3uL7OdinUrFbPur/37G17n3YxZqFRWymwnuzfpxMTGF+vSDZVBg2TvJoOx\ncZ7vZYADtS0r8359RqjomjlTttpYvlwG0H23/dCRkeHs+UTRgpWKCZ1Kxaz7y248BQit+ytQpaIz\npmJVqeh0f5mFil3XFyB7D/kPtuuGyuDBDQfbd++Wr1NnwDw2VqqEf/5TxlPef1/2htJ12WWyjUZC\nguzvRER6GComgp1SbDeeAgTu/gplTMVsK3dfVpXKgQPBVyo6oWI2LqIbKn37SteZEcJffWV+HRQr\nEybIlN4tW+R7M2qUfltA9pF6+WX7PxSIyIuhYiLYKcWhViqhjKkEU6mcOiWva9fWbKDeSaXiHyrf\nfqsXKjExshjRqFY2bbLeWNHMLbfIduL33y/Tern+gqjpMVRM2FUqVgP1oVQqTT2mYjaluKpK7reb\n2tuzZ+Or5zkJld27vQsgT5+W23362LcFZMGhcaWFtWudjYvExcnsq/h42eqbiJqea0PF2PreX12d\n/LPaCh6wnlIcSqWi0/1lVakcOhTclGKdri/AfFqwbqh06iRfs3F1wuJi+XyBQtvX974nW6TU1ckV\nDJ1OtZ02Tbb6tjs/RBQerg2Vs882X6diVCmBukqaqlJp6u4v/0pl/377QXpA9rPyX8BYWiozu3QM\nGOC9SuD27XJb1yWXAPn5clXFlBSpmoio5XJ1qJhVKnbjKYD1QH1Tz/4K9+JH3UolIUEG6n0vxbpn\nj94CRkAuamSMi2zfHvgCW/7OPVdWwV9zjexLRUQtG0PFj914ChDaQH0os7/MKhWl9Ldp8Q8V3Uol\nJkbCx+jCAmSFusXFNhvxnRq8ebPeOhNfr7wiW5fMnu2sHRE1P4aKn6auVAItftTp/vIfqD9+XNZk\n2C3OM5tSrLNGxZCU5F3EWFsrba0uU+tvyBCZDgzIJXn9L7drJyVFtjwxu+wqEbUsDBU/dgsfAetK\n5cQJve6vcG4oqdP1BchWLFVVDbeh1+3+Ahpeb76sTPbQsroGub8xY+T66xUVMlDvZFowEbUuDBU/\ndlu0ANYD9aFWKsGMqeiGytlnS4Xlu/29bvcX0HAbeiddX4BUSSNHAg8/LFUKKw6i6OXavb+sphTr\nVCqBphTbzf6yqlR0NpTs0qXxNVF0QwWQAffKSu/qe519vwxpabKZI6C/qaOv6dPlioRvvOGsHRG1\nLq6uVKymFLfUSsVsXMRJqMTHN1zEWF6uPy7iGyrbtslFsJy44w7p/rr+emftiKh1cXWohFKpWK1T\nacoxlY4dJQh9w1Bn3y+Df6hUVOhdXwRoHCp211w3o7NYkohaN4aKn+aYUhzM5YQBWZDpX63orKY3\nGN1fgGwjv3evfqikpsoA/cmTwVUqROQODBU/OlOKA3V/6Yyp+IfK6dNSfdi1BRqvNzl0KLhKZf9+\nGaPRvU5IbKxUJ/n50m3mdEyFiNyBoeJHp1IJde+v48cbTu01qhSdXXT9KxUna018KxUn4ymGrCzg\ngQdkk0fO4CIiMwwVP6FWKnahEhMj/3xfW6fry+BfqThdwLhnj3wcTKj86Edyed377nPWjojcw7VT\nikOtVIIdqAe8W7UYr6MzSG/w38Orqsr+GvOG1FS53jog4ZKUpNfOMGSI7BZst1U+EbmXa98e2rY1\nn1KsW6kE2/0FNB6sP3pUP1T8dxt2UqkYV2FUSmZyOdnY0cBAIaJAXPsW0RSVis5APSABUl3tva2z\nIaTBv1JxEirnnCNhun9/8KFCRBRIREKlqqoK2dnZGDBgACZMmIBDZhdPB5CXl4f09HSkpaVh4cKF\nZ+7Pzc1FcnIyMjIykJGRgby8PMfHEMqYSkyMTMn13Qoe0K9UunRpuDHkkSNyn464OOnyMjgJFUCu\nuLhrl/PrmhAR6YhIqCxYsADZ2dnYvn07xo8fjwULFjR6Tl1dHWbPno28vDwUFhZi6dKl2Lp1KwDA\n4/HggQcewMaNG7Fx40Zcfvnljo8hlMWPHo95F1iwoeJ0Vfy+ffKxUhIqumMqgPea7998w2nBRBR+\nEQmVFStWYPr06QCA6dOn46233mr0nIKCAvTv3x+pqamIjY3F1KlTsXz58jOPK985uUEIZUNJQMZF\n/ENFd6A+lEolPt47Lbi6Wqb26l6aFwBGjwaef16uHa8744yISFdEQqWyshIJ3+3ZkZCQgErfqz99\np7S0FCk+lxZMTk5GqXFBDwDPPvssRowYgRkzZlh2nwXSti1w6pR0Y/nSqVSAxosYldIfUzGrVHRD\nxXetyb59+htCGiZPBtavB3J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bb7/FVVddhc2bN0f6UIjCjpUKUTPj33EUzRgqREQUNgwV\nombWuXNnHD16NNKHQdQkGCpEzax79+646KKLMGzYMA7UU9ThQD0REYUNKxUiIgobhgoREYUNQ4WI\niMKGoUJERGHDUCEiorBhqBARUdgwVIiIKGwYKkREFDb/Hxu/IkiMyOjQAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f2bf1615f50>"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For a second example, suppose that you want to solve the following\n",
      "coupled, second-order differential equations,\n",
      "$$\n",
      "\\frac{d^2x}{dt^2} = ay\n",
      "$$\n",
      "and\n",
      "$$\n",
      "\\frac{d^2y}{dt^2} = b + c\\frac{dx}{dt}.\n",
      "$$\n",
      "If we make the definitions,\n",
      "$$\n",
      "z[0]=x,\n",
      "$$\n",
      "$$\n",
      "z[1] = \\frac{dx}{dt},\n",
      "$$\n",
      "$$\n",
      "z[2]=y,\n",
      "$$\n",
      "and\n",
      "$$\n",
      "z[3] = \\frac{dy}{dt}.\n",
      "$$\n",
      "then the two second-order equations can be written as the four first-order equations,\n",
      "$$\n",
      "\\frac{dz[0]}{dt} = z[1],\n",
      "$$\n",
      "$$\n",
      "\\frac{dz[1]}{dt} = az[2],\n",
      "$$\n",
      "$$\n",
      "\\frac{dz[2]}{dt} = z[3],\n",
      "$$\n",
      "and\n",
      "$$\n",
      "\\frac{dz[3]}{dt} = b + cz[1].\n",
      "$$\n",
      "These equations are now in a form necessary for the derivative function, w\n",
      "hich would be an array with four elements. Notice that the index of the array is not the number of derivatives of a single function in this case.\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy.integrate import odeint\n",
      "from pylab import *  # for plotting commands\n",
      "\n",
      "def deriv(z,t):  # return derivatives of the array y\n",
      "    a = -0.1\n",
      "    b = 0.2\n",
      "    c = 0.1\n",
      "    return array([z[1], a*z[2], z[3], b+c*z[1]])\n",
      "\n",
      "time = linspace(0.0,10.0,1000)\n",
      "zinit = array([0,0.2,0,0.1])  # initial values\n",
      "z = odeint(deriv,zinit,time)\n",
      "\n",
      "figure()\n",
      "plot(time,z[:,0],label='x')  # z[:,0] is the first column of z\n",
      "plot(time,z[:,2],label='y')  # z[:,2] is the third column of z\n",
      "xlabel('t')\n",
      "legend(loc='upper left')\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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YWGUcRkSkSgwYAGvXmi36efOsrqZ0KqUlP3bsWHbt2oWTkxMtW7Zk4cKF+Pr6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\nAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f2bf3247e10>"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Exercise"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "An example of a differential equation that exhibits chaotic behavior is \n",
      "$$\\frac{d^3x}{dt^3} = -2.017 \\frac{d^2x}{dt^2} + \\left(\\frac{dx}{dt}\\right)^2 - x.$$\n",
      "<ol>\n",
      "<li> Write the differential equation as a set of three <i>first-order</i>\n",
      "differential equations.\n",
      "<li> Modify the example program to solve the equations with the initial conditions of $x=0$, $dx/dt = 0$, and $d^2x/dt^2 = 1$.\n",
      "<li> Plot the results for $x$ for $t$ from 0 to 100.\n",
      "</ol>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Additional Documentation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Further information is available at:  \n",
      "http://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.odeint.html  \n",
      "http://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}