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To solve this equation with `odeint`, we must first convert\nit to a system of first order equations. By defining the angular\nvelocity ``omega(t) = theta'(t)``, we obtain the system::\n\n theta'(t) = omega(t)\n omega'(t) = -b*omega(t) - c*sin(theta(t))\n\nLet `y` be the vector [`theta`, `omega`]. We implement this system\nin Python as:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\ndef pend(y, t, b, c):\n theta, omega = y\n dydt = [omega, -b*omega - c*np.sin(theta)]\n return dydt\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nWe assume the constants are `b` = 0.25 and `c` = 5.0:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "b = 0.25\nc = 5.0\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nFor initial conditions, we assume the pendulum is nearly vertical\nwith `theta(0)` = `pi` - 0.1, and is initially at rest, so\n`omega(0)` = 0. Then the vector of initial conditions is\n\n" }, { "__type": "Code", "__tag": 4050, "value": "y0 = [np.pi - 0.1, 0.0]\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nWe will generate a solution at 101 evenly spaced samples in the interval\n0 <= `t` <= 10. So our array of times is:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "t = np.linspace(0, 10, 101)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nCall `odeint` to generate the solution. To pass the parameters\n`b` and `c` to `pend`, we give them to `odeint` using the `args`\nargument.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy.integrate import odeint\nsol = odeint(pend, y0, t, args=(b, c))\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nThe solution is an array with shape (101, 2). The first column\nis `theta(t)`, and the second is `omega(t)`. 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