{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "a86613cf",
   "metadata": {},
   "outputs": [],
   "source": [
    "using SymPy"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "11fd813c",
   "metadata": {},
   "source": [
    "## 1. Formulas for the Laplace Transform"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "92a5a4d3",
   "metadata": {},
   "source": [
    "Our text book contains tables with the Laplace transforms for most of the common functions.  SymPy offers the functions ``sympy.laplace_transform`` and ``sympy.inverse_laplace_transform`` to compute the Laplace transform and its inverse."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "e2b4961b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(s, t, a, n)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@syms s, t, a::positive, n::integer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "44b91ac3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1/s, 0, True)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lap1 = sympy.laplace_transform(1, t, s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "2f01245d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "Heaviside(t)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "invlap1 = sympy.inverse_laplace_transform(lap1[1], s, t)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8393a406",
   "metadata": {},
   "source": [
    "The $\\theta(t)$ function is the Heaviside function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "51d9e43f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(gamma(n + 1)/(s*s^n), 0, n > -1)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "laptn = sympy.laplace_transform(t^n, t, s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "10bffe74",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "invlaptn = sympy.inverse_laplace_transform(laptn[1], s, t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "4a70ca35",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1/(a + s), -a, True)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lapexp = sympy.laplace_transform(exp(-a*t), t, s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "f623b2bf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-exp(-a*t)*Ei(-a*t), 0, True)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "invlapexp = sympy.laplace_transform(lapexp[1], s, t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "db76fcf5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(s/(a^2 + s^2), 0, True)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lapcos = sympy.laplace_transform(cos(a*t), t, s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "307a916e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\cos{\\left(a t \\right)} \\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "cos(a*t)*Heaviside(t)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "invlapcos = sympy.inverse_laplace_transform(lapcos[1], s, t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "dfbae889",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(a/(a^2 + s^2), 0, True)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lapsin = sympy.laplace_transform(sin(a*t), t, s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "636e5429",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\sin{\\left(a t \\right)} \\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "sin(a*t)*Heaviside(t)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "invlapsin = sympy.inverse_laplace_transform(lapsin[1], s, t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "9c3ebd42",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(s/(-a^2 + s^2), a, True)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lapcosh = sympy.laplace_transform(cosh(a*t), t, s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "24d7e399",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\frac{e^{a t} \\theta\\left(t\\right)}{2} + \\frac{e^{- a t} \\theta\\left(t\\right)}{2}$"
      ],
      "text/plain": [
       " a*t                 -a*t             \n",
       "e   *Heaviside(t)   e    *Heaviside(t)\n",
       "----------------- + ------------------\n",
       "        2                   2         "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "invlapcosh = sympy.inverse_laplace_transform(lapcosh[1], s, t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "3168fb93",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(a/(-a^2 + s^2), a, True)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lapsinh = sympy.laplace_transform(sinh(a*t), t, s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "a5fbe1d2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$a \\left(\\frac{e^{a t} \\theta\\left(t\\right)}{2 a} - \\frac{e^{- a t} \\theta\\left(t\\right)}{2 a}\\right)$"
      ],
      "text/plain": [
       "  / a*t                 -a*t             \\\n",
       "  |e   *Heaviside(t)   e    *Heaviside(t)|\n",
       "a*|----------------- - ------------------|\n",
       "  \\       2*a                 2*a        /"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "invlapsinh = sympy.inverse_laplace_transform(lapsinh[1], s, t)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ac31934d",
   "metadata": {},
   "source": [
    "The Laplace transform of the Dirac delta function is 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "048d1353",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\delta\\left(t\\right)$"
      ],
      "text/plain": [
       "DiracDelta(t)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diracdelta = sympy.inverse_laplace_transform(1, s, t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "4451f758",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1, -oo, True)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.laplace_transform(diracdelta, t, s)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b59f4bbd",
   "metadata": {},
   "source": [
    "## 2. Solving a System of Differential equations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3588398c",
   "metadata": {},
   "source": [
    "The Laplace transform of\n",
    "\n",
    "$$\n",
    "    \\left\\{\n",
    "       \\begin{array}{rcr}\n",
    "          x' & = & -2 x - 9 y \\\\\n",
    "          y' & = &    x - 2 y\n",
    "       \\end{array}\n",
    "    \\right. \\quad x(0) = -1, y(0) = 1\n",
    "$$\n",
    "\n",
    "is\n",
    "\n",
    "$$\n",
    "    \\left\\{\n",
    "       \\begin{array}{rcr}\n",
    "          s X + 1 & = & -2 X - 9 Y \\\\\n",
    "          s Y - 1 & = &    X - 2 Y\n",
    "       \\end{array}\n",
    "    \\right.\n",
    "$$\n",
    "\n",
    "or\n",
    "\n",
    "$$\n",
    "    \\left\\{\n",
    "       \\begin{array}{rcr}\n",
    "          (s+2) X + 9 Y & = & -1 \\\\\n",
    "          -X + (s+2) Y & = & 1\n",
    "       \\end{array}\n",
    "    \\right.\n",
    "$$\n",
    "\n",
    "where $X(s)$ and $Y(s)$ are the Laplace transforms\n",
    "of $x(t)$ and $y(t)$ respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "be624f75",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(X, Y)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@syms X, Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "33c55e17",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left[ \\begin{array}{r}X \\left(s + 2\\right) + 9 Y + 1\\\\- X + Y \\left(s + 2\\right) - 1\\end{array} \\right]$\n"
      ],
      "text/plain": [
       "2-element Vector{Sym}:\n",
       " X*(s + 2) + 9*Y + 1\n",
       "  -X + Y*(s + 2) - 1"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq1 = (s+2)*X + 9*Y + 1\n",
    "eq2 = -X + (s+2)*Y - 1\n",
    "eqs = [eq1, eq2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "e4df4480",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{Any, Any} with 2 entries:\n",
       "  X => (-s - 11)/(s^2 + 4*s + 13)\n",
       "  Y => (s + 1)/(s^2 + 4*s + 13)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol = sympy.solve(eqs, [X, Y])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "75f4b7cf",
   "metadata": {},
   "source": [
    "Now we transform the solution into the type domain."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "f9b3ef11",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left(- 3 e^{- 2 t} \\sin{\\left(3 t \\right)} - e^{- 2 t} \\cos{\\left(3 t \\right)}\\right) \\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "/     -2*t             -2*t         \\             \n",
       "\\- 3*e    *sin(3*t) - e    *cos(3*t)/*Heaviside(t)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xt = sympy.inverse_laplace_transform(sol[X], s, t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "fe00458a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\left(- \\frac{e^{- 2 t} \\sin{\\left(3 t \\right)}}{3} + e^{- 2 t} \\cos{\\left(3 t \\right)}\\right) \\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "/   -2*t                          \\             \n",
       "|  e    *sin(3*t)    -2*t         |             \n",
       "|- -------------- + e    *cos(3*t)|*Heaviside(t)\n",
       "\\        3                        /             "
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "yt = sympy.inverse_laplace_transform(sol[Y], s, t)"
   ]
  }
 ],
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