{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "fd6e1fdd",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.integrate import solve_ivp\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "551af545",
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import pi, exp\n",
    "def f(t,y):\n",
    "    return np.exp(-(t**2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "id": "b79ae989",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Integral')"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(0,10,100)\n",
    "soln = solve_ivp(f, (0,10), (0,), t_eval = x)\n",
    "fig, ax = plt.subplots(1,1)\n",
    "ax.plot(x, f(x,0)) \n",
    "ax.plot(soln.t, soln.y[0])\n",
    "ax.set_title(\"Integral\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "id": "412b69d2",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4990788239579371"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2*soln.y[0][-1]**2/pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "id": "879f9330",
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(t,y):\n",
    "    return np.exp(-2*(t**2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "94589455",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Integral')"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(0,10,100)\n",
    "soln = solve_ivp(f, (0,10), (0,), t_eval = x)\n",
    "fig, ax = plt.subplots(1,1)\n",
    "ax.plot(x, f(x,0)) \n",
    "ax.plot(soln.t, soln.y[0])\n",
    "ax.set_title(\"Integral\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "id": "4f5fca28",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.24979989298663194"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2*soln.y[0][-1]**2/pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "id": "ef3b9541",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - 4 t e^{- 2 t^{2}}$"
      ],
      "text/plain": [
       "-4*t*exp(-2*t**2)"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sym\n",
    "from sympy.abc import t as ts\n",
    "#sym.diff(sym.exp())\n",
    "sym.diff(sym.exp(-2*(ts**2)), ts)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "37b23981",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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