{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "1492f1f5",
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sym "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "f4d5f61f",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sym.Symbol(\"x\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "010c42af",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f{\\left(x \\right)}$"
      ],
      "text/plain": [
       "f(x)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = sym.Function('f')(x)\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "f47319a5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{d}{d x} f{\\left(x \\right)}$"
      ],
      "text/plain": [
       "Derivative(f(x), x)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.diff()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "628ae0b5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} \\frac{d^{2}}{d x^{2}} f{\\left(x \\right)} + \\frac{d}{d x} f{\\left(x \\right)} = 1$"
      ],
      "text/plain": [
       "Eq(x**2*Derivative(f(x), (x, 2)) + Derivative(f(x), x), 1)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff_eq = sym.Eq(x**2*f.diff(x,x)+f.diff() , 1)\n",
    "diff_eq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "39b034e0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff_eq.rhs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "62fe83b7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} \\frac{d^{2}}{d x^{2}} f{\\left(x \\right)} + \\frac{d}{d x} f{\\left(x \\right)}$"
      ],
      "text/plain": [
       "x**2*Derivative(f(x), (x, 2)) + Derivative(f(x), x)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff_eq.lhs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "74c4593c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f{\\left(x \\right)} = C_{1} + C_{2} \\left(x e^{\\frac{1}{x}} - \\operatorname{Ei}{\\left(\\frac{1}{x} \\right)}\\right) + x$"
      ],
      "text/plain": [
       "Eq(f(x), C1 + C2*(x*exp(1/x) - Ei(1/x)) + x)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol = sym.dsolve(diff_eq, f)\n",
    "sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "9a0702fe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.relational.Equality"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(sol)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "88d17b44",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle C_{1} + C_{2} \\log{\\left(x \\right)} + \\frac{x^{4}}{16}$"
      ],
      "text/plain": [
       "C1 + C2*log(x) + x**4/16"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp = sol.rhs\n",
    "exp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "313d30f6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[C1, C2]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c2,_, c1 = tuple(exp.free_symbols)\n",
    "[c1, c2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "43a91997",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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