calculus_-_differential equations.ipynb

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  {
   "source": [
    "# Differential equations with Python"
   ],
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   "source": [
    "import numpy as np \n",
    "from scipy.integrate import odeint\n",
    "import matplotlib.pyplot as plt \n",
    "%matplotlib inline\n"
   ]
  },
  {
   "source": [
    "### Example 1\n",
    "$\n",
    "\\frac{dy(t)}{dt} = - k y(t)\n",
    "$"
   ],
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   "source": [
    "def model(y,t, k):\n",
    "    dydt = -k * y\n",
    "    return dydt"
   ]
  },
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   "execution_count": 13,
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "y0 = 5\n",
    "t = np.linspace(0, 20)\n",
    "k= 0.4\n",
    "y1 = odeint(model, y0, t, args=(k,))\n",
    "k= 0.5\n",
    "y2 = odeint(model, y0, t, args=(k,))\n",
    "k= 0.2\n",
    "y3 = odeint(model, y0, t, args=(k,))\n",
    "\n",
    "plt.plot(t, y1)\n",
    "plt.plot(t, y2)\n",
    "plt.plot(t, y3)\n",
    "plt.xlabel('time')\n",
    "plt.ylabel('y(t)')"
   ]
  },
  {
   "source": [
    "## Example 2\n",
    "\n",
    "$$\n",
    " y'' + \\omega^2 y = 0\n",
    "$$\n",
    "\n",
    "Write the second order differential equation as a system of first order differential equations:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "x_1(t) &= y(t) \\\\\n",
    "x_2(t) &= y'(t) \\\\\n",
    "\\end{align}\n",
    "$$\n",
    "differentiate both sides:\n",
    "$$\n",
    "\\begin{align}\n",
    "x'_1 &= y' = x_2\\\\\n",
    "x'_2 &= y'' = -\\omega^2  y'=-\\omega^2 x_1\\\\\n",
    "\\end{align}\n",
    "$$\n",
    "\n"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "def model_2(y, t, omega):\n",
    "    dydt1 = y[1]\n",
    "    dydt2= - omega**2 * y[0]\n",
    "    return [dydt1, dydt2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x203026d4640>]"
      ]
     },
     "metadata": {},
     "execution_count": 34
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "<Figure size 432x288 with 1 Axes>",
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "plt.plot(t, sol[:,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ]
}