HajimeKawahara/exojax

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "efd6143b",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import sympy\n",
    "from sympy import exp\n",
    "\n",
    "t = sympy.Symbol('t')\n",
    "twp = sympy.Symbol('twp')\n",
    "tref = sympy.Symbol('tref')\n",
    "E = sympy.Symbol('E')\n",
    "c2 = sympy.Symbol('c2')\n",
    "p = sympy.Symbol('p')\n",
    "dE = sympy.Symbol('dE')\n",
    "El = sympy.Symbol('El')\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "515e621e",
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(E,t):\n",
    "    return exp(- c2*E*(t-tref))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "c6e796f9",
   "metadata": {},
   "outputs": [],
   "source": [
    "E1 = El - p*dE\n",
    "E2 = El + (1-p)*dE\n",
    "xl = f(El, twp)\n",
    "x1 = f(E1, twp)\n",
    "x2 = f(E2, twp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "ada4205a",
   "metadata": {},
   "outputs": [],
   "source": [
    "xpl = sympy.diff(xl,twp)\n",
    "xp1 = sympy.diff(x1,twp)\n",
    "xp2 = sympy.diff(x2,twp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "a3a0dc8a",
   "metadata": {},
   "outputs": [],
   "source": [
    "w1 = (x2 - xl)/(x2 - x1)\n",
    "w2 = (xl - x1)/(x2 - x1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "f24f64f6",
   "metadata": {},
   "outputs": [],
   "source": [
    "wp1 = sympy.diff(w1,twp)\n",
    "wp2 = sympy.diff(w2,twp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "ba18354c",
   "metadata": {},
   "outputs": [],
   "source": [
    "Stilde = w1*f(E1,t)+w2*f(E2,t) + f(E1,t)*wp1*(t - tref) + f(E2,t)*wp2*(t - tref)\n",
    "S = f(El,t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "dce54f7a",
   "metadata": {},
   "outputs": [],
   "source": [
    "dq=sympy.simplify(sympy.diff(Stilde - S,p))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "c20e358c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{c_{2} dE \\left(c_{2} \\left(t - tref\\right)^{2} \\left(\\left(El e^{El c_{2} \\left(tref - twp\\right)} + \\left(- El + dE p\\right) e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)}\\right) \\left(e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) + \\left(\\left(- El + dE \\left(p - 1\\right)\\right) e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)} + \\left(El - dE p\\right) e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)}\\right) \\left(- e^{El c_{2} \\left(tref - twp\\right)} + e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)}\\right)\\right) e^{c_{2} \\left(t - tref\\right) \\left(7 El - 4 dE p - 3 dE \\left(p - 1\\right)\\right)} - c_{2} \\left(t - tref\\right)^{2} \\left(\\left(El e^{El c_{2} \\left(tref - twp\\right)} + \\left(- El + dE \\left(p - 1\\right)\\right) e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) \\left(e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) - \\left(\\left(- El + dE \\left(p - 1\\right)\\right) e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)} + \\left(El - dE p\\right) e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)}\\right) \\left(e^{El c_{2} \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right)\\right) e^{c_{2} \\left(t - tref\\right) \\left(7 El - 3 dE p - 4 dE \\left(p - 1\\right)\\right)} + \\left(t - tref\\right) \\left(c_{2} \\left(tref - twp\\right) \\left(El e^{El c_{2} \\left(tref - twp\\right)} + \\left(- El + dE p\\right) e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)}\\right) \\left(e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) - 2 c_{2} \\left(tref - twp\\right) \\left(\\left(- El + dE \\left(p - 1\\right)\\right) e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)} + \\left(El - dE p\\right) e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)}\\right) \\left(e^{El c_{2} \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)}\\right) - c_{2} \\left(tref - twp\\right) \\left(\\left(- El + dE \\left(p - 1\\right)\\right) e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)} + \\left(El - dE p\\right) e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)}\\right) e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} + \\left(c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right) + 1\\right) \\left(e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} + \\left(e^{El c_{2} \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)}\\right) \\left(c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right) e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} - c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right) e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)} + e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right)\\right) e^{c_{2} \\left(t - tref\\right) \\left(7 El - 4 dE p - 3 dE \\left(p - 1\\right)\\right)} + \\left(t - tref\\right) \\left(- c_{2} \\left(tref - twp\\right) \\left(El e^{El c_{2} \\left(tref - twp\\right)} + \\left(- El + dE \\left(p - 1\\right)\\right) e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) \\left(e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) + 2 c_{2} \\left(tref - twp\\right) \\left(\\left(- El + dE \\left(p - 1\\right)\\right) e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)} + \\left(El - dE p\\right) e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)}\\right) \\left(e^{El c_{2} \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) + c_{2} \\left(tref - twp\\right) \\left(\\left(- El + dE \\left(p - 1\\right)\\right) e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)} + \\left(El - dE p\\right) e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)}\\right) e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)} - \\left(c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right) + 1\\right) \\left(e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)} + \\left(e^{El c_{2} \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) \\left(- c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right) e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} + c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right) e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} + e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right)\\right) e^{c_{2} \\left(t - tref\\right) \\left(7 El - 3 dE p - 4 dE \\left(p - 1\\right)\\right)} - \\left(tref - twp\\right) \\left(e^{El c_{2} \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)}\\right) \\left(e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) e^{c_{2} \\left(t - tref\\right) \\left(7 El - 4 dE p - 3 dE \\left(p - 1\\right)\\right)} + \\left(tref - twp\\right) \\left(e^{El c_{2} \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) \\left(e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) e^{c_{2} \\left(t - tref\\right) \\left(7 El - 3 dE p - 4 dE \\left(p - 1\\right)\\right)} + \\left(- \\left(t - tref\\right) \\left(e^{El c_{2} \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)}\\right) + \\left(- tref + twp\\right) e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)}\\right) \\left(e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) e^{c_{2} \\left(t - tref\\right) \\left(7 El - 4 dE p - 3 dE \\left(p - 1\\right)\\right)} + \\left(\\left(t - tref\\right) \\left(e^{El c_{2} \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) + \\left(tref - twp\\right) e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) \\left(e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right) e^{c_{2} \\left(t - tref\\right) \\left(7 El - 3 dE p - 4 dE \\left(p - 1\\right)\\right)}\\right) e^{4 c_{2} \\left(t - tref\\right) \\left(- 2 El + dE p + dE \\left(p - 1\\right)\\right)}}{\\left(e^{c_{2} \\left(El - dE p\\right) \\left(tref - twp\\right)} - e^{c_{2} \\left(El - dE \\left(p - 1\\right)\\right) \\left(tref - twp\\right)}\\right)^{2}}$"
      ],
      "text/plain": [
       "c2*dE*(c2*(t - tref)**2*((El*exp(El*c2*(tref - twp)) + (-El + dE*p)*exp(c2*(El - dE*p)*(tref - twp)))*(exp(c2*(El - dE*p)*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp))) + ((-El + dE*(p - 1))*exp(c2*(El - dE*(p - 1))*(tref - twp)) + (El - dE*p)*exp(c2*(El - dE*p)*(tref - twp)))*(-exp(El*c2*(tref - twp)) + exp(c2*(El - dE*p)*(tref - twp))))*exp(c2*(t - tref)*(7*El - 4*dE*p - 3*dE*(p - 1))) - c2*(t - tref)**2*((El*exp(El*c2*(tref - twp)) + (-El + dE*(p - 1))*exp(c2*(El - dE*(p - 1))*(tref - twp)))*(exp(c2*(El - dE*p)*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp))) - ((-El + dE*(p - 1))*exp(c2*(El - dE*(p - 1))*(tref - twp)) + (El - dE*p)*exp(c2*(El - dE*p)*(tref - twp)))*(exp(El*c2*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp))))*exp(c2*(t - tref)*(7*El - 3*dE*p - 4*dE*(p - 1))) + (t - tref)*(c2*(tref - twp)*(El*exp(El*c2*(tref - twp)) + (-El + dE*p)*exp(c2*(El - dE*p)*(tref - twp)))*(exp(c2*(El - dE*p)*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp))) - 2*c2*(tref - twp)*((-El + dE*(p - 1))*exp(c2*(El - dE*(p - 1))*(tref - twp)) + (El - dE*p)*exp(c2*(El - dE*p)*(tref - twp)))*(exp(El*c2*(tref - twp)) - exp(c2*(El - dE*p)*(tref - twp))) - c2*(tref - twp)*((-El + dE*(p - 1))*exp(c2*(El - dE*(p - 1))*(tref - twp)) + (El - dE*p)*exp(c2*(El - dE*p)*(tref - twp)))*exp(c2*(El - dE*p)*(tref - twp)) + (c2*(El - dE*p)*(tref - twp) + 1)*(exp(c2*(El - dE*p)*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp)))*exp(c2*(El - dE*p)*(tref - twp)) + (exp(El*c2*(tref - twp)) - exp(c2*(El - dE*p)*(tref - twp)))*(c2*(El - dE*p)*(tref - twp)*exp(c2*(El - dE*p)*(tref - twp)) - c2*(El - dE*(p - 1))*(tref - twp)*exp(c2*(El - dE*(p - 1))*(tref - twp)) + exp(c2*(El - dE*p)*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp))))*exp(c2*(t - tref)*(7*El - 4*dE*p - 3*dE*(p - 1))) + (t - tref)*(-c2*(tref - twp)*(El*exp(El*c2*(tref - twp)) + (-El + dE*(p - 1))*exp(c2*(El - dE*(p - 1))*(tref - twp)))*(exp(c2*(El - dE*p)*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp))) + 2*c2*(tref - twp)*((-El + dE*(p - 1))*exp(c2*(El - dE*(p - 1))*(tref - twp)) + (El - dE*p)*exp(c2*(El - dE*p)*(tref - twp)))*(exp(El*c2*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp))) + c2*(tref - twp)*((-El + dE*(p - 1))*exp(c2*(El - dE*(p - 1))*(tref - twp)) + (El - dE*p)*exp(c2*(El - dE*p)*(tref - twp)))*exp(c2*(El - dE*(p - 1))*(tref - twp)) - (c2*(El - dE*(p - 1))*(tref - twp) + 1)*(exp(c2*(El - dE*p)*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp)))*exp(c2*(El - dE*(p - 1))*(tref - twp)) + (exp(El*c2*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp)))*(-c2*(El - dE*p)*(tref - twp)*exp(c2*(El - dE*p)*(tref - twp)) + c2*(El - dE*(p - 1))*(tref - twp)*exp(c2*(El - dE*(p - 1))*(tref - twp)) - exp(c2*(El - dE*p)*(tref - twp)) + exp(c2*(El - dE*(p - 1))*(tref - twp))))*exp(c2*(t - tref)*(7*El - 3*dE*p - 4*dE*(p - 1))) - (tref - twp)*(exp(El*c2*(tref - twp)) - exp(c2*(El - dE*p)*(tref - twp)))*(exp(c2*(El - dE*p)*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp)))*exp(c2*(t - tref)*(7*El - 4*dE*p - 3*dE*(p - 1))) + (tref - twp)*(exp(El*c2*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp)))*(exp(c2*(El - dE*p)*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp)))*exp(c2*(t - tref)*(7*El - 3*dE*p - 4*dE*(p - 1))) + (-(t - tref)*(exp(El*c2*(tref - twp)) - exp(c2*(El - dE*p)*(tref - twp))) + (-tref + twp)*exp(c2*(El - dE*p)*(tref - twp)))*(exp(c2*(El - dE*p)*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp)))*exp(c2*(t - tref)*(7*El - 4*dE*p - 3*dE*(p - 1))) + ((t - tref)*(exp(El*c2*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp))) + (tref - twp)*exp(c2*(El - dE*(p - 1))*(tref - twp)))*(exp(c2*(El - dE*p)*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp)))*exp(c2*(t - tref)*(7*El - 3*dE*p - 4*dE*(p - 1))))*exp(4*c2*(t - tref)*(-2*El + dE*p + dE*(p - 1)))/(exp(c2*(El - dE*p)*(tref - twp)) - exp(c2*(El - dE*(p - 1))*(tref - twp)))**2"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "be786709",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{c_{2} dE \\left(t e^{c_{2} dE \\left(p tref + t + twp\\right)} - t e^{c_{2} dE \\left(p tref + tref + twp\\right)} - t e^{c_{2} dE \\left(p twp + t + tref\\right)} + t e^{c_{2} dE \\left(p twp + tref + twp\\right)} + tref e^{c_{2} dE \\left(p twp + t + tref\\right)} - tref e^{c_{2} dE \\left(p twp + tref + twp\\right)} - twp e^{c_{2} dE \\left(p tref + t + twp\\right)} + twp e^{c_{2} dE \\left(p tref + tref + twp\\right)}\\right) e^{- c_{2} \\left(El t - El tref - dE p t + dE p tref + dE p twp + dE t\\right)}}{e^{c_{2} dE tref} - e^{c_{2} dE twp}}$"
      ],
      "text/plain": [
       "-c2*dE*(t*exp(c2*dE*(p*tref + t + twp)) - t*exp(c2*dE*(p*tref + tref + twp)) - t*exp(c2*dE*(p*twp + t + tref)) + t*exp(c2*dE*(p*twp + tref + twp)) + tref*exp(c2*dE*(p*twp + t + tref)) - tref*exp(c2*dE*(p*twp + tref + twp)) - twp*exp(c2*dE*(p*tref + t + twp)) + twp*exp(c2*dE*(p*tref + tref + twp)))*exp(-c2*(El*t - El*tref - dE*p*t + dE*p*tref + dE*p*twp + dE*t))/(exp(c2*dE*tref) - exp(c2*dE*twp))"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "ww= sympy.simplify(dq)\n",
    "ww"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "135dd1f7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{c_{2} dE \\left(t e^{c_{2} dE \\left(tref + 1.5 twp\\right)} - t e^{c_{2} dE \\left(1.5 tref + twp\\right)} + t e^{c_{2} dE \\left(t + 0.5 tref + twp\\right)} - t e^{c_{2} dE \\left(t + tref + 0.5 twp\\right)} - tref e^{c_{2} dE \\left(tref + 1.5 twp\\right)} + tref e^{c_{2} dE \\left(t + tref + 0.5 twp\\right)} + twp e^{c_{2} dE \\left(1.5 tref + twp\\right)} - twp e^{c_{2} dE \\left(t + 0.5 tref + twp\\right)}\\right) e^{- c_{2} \\left(El t - El tref + 0.5 dE t + 0.5 dE tref + 0.5 dE twp\\right)}}{e^{c_{2} dE tref} - e^{c_{2} dE twp}}$"
      ],
      "text/plain": [
       "-c2*dE*(t*exp(c2*dE*(tref + 1.5*twp)) - t*exp(c2*dE*(1.5*tref + twp)) + t*exp(c2*dE*(t + 0.5*tref + twp)) - t*exp(c2*dE*(t + tref + 0.5*twp)) - tref*exp(c2*dE*(tref + 1.5*twp)) + tref*exp(c2*dE*(t + tref + 0.5*twp)) + twp*exp(c2*dE*(1.5*tref + twp)) - twp*exp(c2*dE*(t + 0.5*tref + twp)))*exp(-c2*(El*t - El*tref + 0.5*dE*t + 0.5*dE*tref + 0.5*dE*twp))/(exp(c2*dE*tref) - exp(c2*dE*twp))"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.simplify(ww.subs([(p,1/2)]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "bffe59dd",
   "metadata": {},
   "outputs": [
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[0;32m/tmp/ipykernel_4558/1805557983.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mans\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msolve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mEq\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdq\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/sympy/solvers/solvers.py\u001b[0m in \u001b[0;36msolve\u001b[0;34m(f, *symbols, **flags)\u001b[0m\n\u001b[1;32m   1093\u001b[0m     \u001b[0;31m###########################################################################\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1094\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mbare_f\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1095\u001b[0;31m         \u001b[0msolution\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_solve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0msymbols\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mflags\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1096\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1097\u001b[0m         \u001b[0msolution\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_solve_system\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msymbols\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mflags\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/sympy/solvers/solvers.py\u001b[0m in \u001b[0;36m_solve\u001b[0;34m(f, *symbols, **flags)\u001b[0m\n\u001b[1;32m   1704\u001b[0m         \u001b[0mflags\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpop\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'tsolve'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m)\u001b[0m  \u001b[0;31m# allow tsolve to be used on next pass\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1705\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1706\u001b[0;31m             \u001b[0msoln\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_tsolve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf_num\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msymbol\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mflags\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1707\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0msoln\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1708\u001b[0m                 \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msoln\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/sympy/solvers/solvers.py\u001b[0m in \u001b[0;36m_tsolve\u001b[0;34m(eq, sym, **flags)\u001b[0m\n\u001b[1;32m   2743\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mpos\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mhas\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mu\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2744\u001b[0m             \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2745\u001b[0;31m                 \u001b[0msoln\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_solve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpos\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mflags\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2746\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mlist\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mordered\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mreps\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ms\u001b[0m \u001b[0;32min\u001b[0m \u001b[0msoln\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2747\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mNotImplementedError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/sympy/solvers/solvers.py\u001b[0m in \u001b[0;36m_solve\u001b[0;34m(f, *symbols, **flags)\u001b[0m\n\u001b[1;32m   1381\u001b[0m                 \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mset\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1382\u001b[0m                 \u001b[0;32mbreak\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1383\u001b[0;31m             \u001b[0msoln\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_solve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mm\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msymbol\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mflags\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1384\u001b[0m             \u001b[0mresult\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mupdate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mset\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msoln\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1385\u001b[0m         \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlist\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresult\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/sympy/solvers/solvers.py\u001b[0m in \u001b[0;36m_solve\u001b[0;34m(f, *symbols, **flags)\u001b[0m\n\u001b[1;32m   1715\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1716\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mflags\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'simplify'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1717\u001b[0;31m         \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlist\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmap\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msimplify\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1718\u001b[0m         \u001b[0;31m# we just simplified the solution so we now set the flag to\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1719\u001b[0m         \u001b[0;31m# False so the simplification doesn't happen again in checksol()\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/sympy/simplify/simplify.py\u001b[0m in \u001b[0;36msimplify\u001b[0;34m(expr, ratio, measure, rational, inverse, doit, **kwargs)\u001b[0m\n\u001b[1;32m    721\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    722\u001b[0m     \u001b[0mshort\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mshorter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpowsimp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcombine\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m'exp'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdeep\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mpowsimp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 723\u001b[0;31m     \u001b[0mshort\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mshorter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mshort\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcancel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mshort\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    724\u001b[0m     \u001b[0mshort\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mshorter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mshort\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfactor_terms\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mshort\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexpand_power_exp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpand_mul\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mshort\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    725\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mshort\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mhas\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mTrigonometricFunction\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mHyperbolicFunction\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mExpBase\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/sympy/simplify/simplify.py\u001b[0m in \u001b[0;36mshorter\u001b[0;34m(*choices)\u001b[0m\n\u001b[1;32m    571\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mhas_variety\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mchoices\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    572\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mchoices\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 573\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mmin\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mchoices\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkey\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mmeasure\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    574\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    575\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mdone\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0me\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/sympy/core/function.py\u001b[0m in \u001b[0;36mcount_ops\u001b[0;34m(expr, visual)\u001b[0m\n\u001b[1;32m   3221\u001b[0m                 \u001b[0mo\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mSymbol\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__name__\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mupper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3222\u001b[0m                 \u001b[0;31m# count the args\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3223\u001b[0;31m                 \u001b[0mops\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mo\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3224\u001b[0m             elif a.args and (\n\u001b[1;32m   3225\u001b[0m                     \u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Pow\u001b[0m \u001b[0;32mor\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/sympy/core/decorators.py\u001b[0m in \u001b[0;36m_func\u001b[0;34m(self, other)\u001b[0m\n\u001b[1;32m    264\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mother\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexpectedcls\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    265\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mretval\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 266\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mother\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    267\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    268\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0m_func\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.8/site-packages/sympy/core/decorators.py\u001b[0m in \u001b[0;36mbinary_op_wrapper\u001b[0;34m(self, other)\u001b[0m\n\u001b[1;32m    134\u001b[0m                     \u001b[0;32mif\u001b[0m \u001b[0mf\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    135\u001b[0m                         \u001b[0;32mreturn\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 136\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mother\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    137\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mbinary_op_wrapper\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    138\u001b[0m     \u001b[0;32mreturn\u001b[0m \u001b[0mpriority_decorator\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
     ]
    }
   ],
   "source": [
    "ans = sympy.solve(sympy.Eq(dq,0),p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "fac86abd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(c2*dE*tref - c2*dE*twp - log((t*exp(c2*dE*t) - t*exp(c2*dE*tref) - twp*exp(c2*dE*t) + twp*exp(c2*dE*tref))/(t*exp(c2*dE*t) - t*exp(c2*dE*twp) - tref*exp(c2*dE*t) + tref*exp(c2*dE*twp))))/(c2*dE*(tref - twp))]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ans"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "a36a4594",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{c_{2} dE \\left(t e^{c_{2} dE \\left(p tref + t + twp\\right)} - t e^{c_{2} dE \\left(p tref + tref + twp\\right)} - t e^{c_{2} dE \\left(p twp + t + tref\\right)} + t e^{c_{2} dE \\left(p twp + tref + twp\\right)} + tref e^{c_{2} dE \\left(p twp + t + tref\\right)} - tref e^{c_{2} dE \\left(p twp + tref + twp\\right)} - twp e^{c_{2} dE \\left(p tref + t + twp\\right)} + twp e^{c_{2} dE \\left(p tref + tref + twp\\right)}\\right) e^{- c_{2} \\left(El t - El tref - dE p t + dE p tref + dE p twp + dE t\\right)}}{e^{c_{2} dE tref} - e^{c_{2} dE twp}}$"
      ],
      "text/plain": [
       "-c2*dE*(t*exp(c2*dE*(p*tref + t + twp)) - t*exp(c2*dE*(p*tref + tref + twp)) - t*exp(c2*dE*(p*twp + t + tref)) + t*exp(c2*dE*(p*twp + tref + twp)) + tref*exp(c2*dE*(p*twp + t + tref)) - tref*exp(c2*dE*(p*twp + tref + twp)) - twp*exp(c2*dE*(p*tref + t + twp)) + twp*exp(c2*dE*(p*tref + tref + twp)))*exp(-c2*(El*t - El*tref - dE*p*t + dE*p*tref + dE*p*twp + dE*t))/(exp(c2*dE*tref) - exp(c2*dE*twp))"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ww"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "03631072",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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