6. Laplace.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 6. Laplace Transformation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<small>Sympy scheint [Problemen](https://github.com/sympy/sympy/issues/10354) zu haben mit Laplace und Piecewise, deshalb mache ich das jetzt in Sage.</small>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sage import *\n",
    "%display latex\n",
    "\n",
    "t,s,u=var('t s u')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6.1 Transformation einer stückweise definiertes Funktion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}piecewise\\left(\\left(\\left(\\verb|(0,|\\phantom{\\verb!x!}\\verb|2)|, -t\\right), \\left(\\verb|(2,|\\phantom{\\verb!x!}\\verb|+oo)|, t\\right)\\right), t\\right)</script></html>"
      ],
      "text/plain": [
       "piecewise(t|-->-t on (0, 2), t|-->t on (2, +oo); t)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ff=piecewise([\n",
    "    ((0,2), -t),\n",
    "    ((2,+oo), t),\n",
    "])\n",
    "ff"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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QdsWLmzdTtGhh3kxB2AEAgGSWFEFXoCDs3nlHWrxYKlHCnNfur39N7LAbO3as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\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(ff,0,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(2 \\, s + 1\\right)} e^{\\left(-2 \\, s\\right)}}{s^{2}} - \\frac{1}{s^{2}}</script></html>"
      ],
      "text/plain": [
       "2*(2*s + 1)*e^(-2*s)/s^2 - 1/s^2"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ff.laplace(t,s)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Von der Transformation, es ist möglich die funktion zu schätzen. Hier, zB, $t=0; t=2$ sind wichtig."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6.2 Partialbruchzerlegung und Inverse transformation\n",
    "### 6.2.1 Beispiel 1\n",
    "\"Finde die Funktion dessen Laplace Transform ist...\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy as ss"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{s^{2} - 2 \\, s}{{\\left(s^{2} - 16\\right)} s^{2}}</script></html>"
      ],
      "text/plain": [
       "(s^2 - 2*s)/((s^2 - 16)*s^2)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fpa=(s**2-2*s)/((s**2)*(s**2-16))\n",
    "fpa\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "fpaa=(s**2-2*s)\n",
    "fpab=(s**2)*(s**2-16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}s^{4} - 16 \\, s^{2}</script></html>"
      ],
      "text/plain": [
       "s^4 - 16*s^2"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fpab=fpab.expand()\n",
    "fpab"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{s^{2} - 2 \\, s}{s^{4} - 16 \\, s^{2}}</script></html>"
      ],
      "text/plain": [
       "(s^2 - 2*s)/(s^4 - 16*s^2)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fpa=fpaa/fpab\n",
    "fpa"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Partialbruchzerlegung:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-3/16/(s + 4) + 1/16/(s - 4) + 1/8/s\n"
     ]
    }
   ],
   "source": [
    "print(fpa.partial_fraction(s))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Von jedes Teil:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "fpa1=-3/16/(s + 4)\n",
    "fpa2= 1/16/(s - 4)\n",
    "fpa3= 1/8/s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{3}{16} \\, e^{\\left(-4 \\, t\\right)}</script></html>"
      ],
      "text/plain": [
       "-3/16*e^(-4*t)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i1=inverse_laplace(fpa1,s,t);i1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{16} \\, e^{\\left(4 \\, t\\right)}</script></html>"
      ],
      "text/plain": [
       "1/16*e^(4*t)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i2=inverse_laplace(fpa2,s,t);i2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{8}</script></html>"
      ],
      "text/plain": [
       "1/8"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i3=inverse_laplace(fpa3,s,t);i3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bool(inverse_laplace(fpa,s,t)==i1+i2+i3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 6.2.2 Beispiel 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{2 \\, {\\left(s^{3} + 3 \\, s^{2} - 2 \\, s - 7\\right)}}{s^{4} + 2 \\, s^{3} - 2 \\, s^{2} - 6 \\, s + 5}</script></html>"
      ],
      "text/plain": [
       "2*(s^3 + 3*s^2 - 2*s - 7)/(s^4 + 2*s^3 - 2*s^2 - 6*s + 5)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fpal=(2*s**3+6*s**2-4*s-14)/(s**4+2*s**3-2*s**2-6*s+5); fpal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{s^{2} + 4 \\, s + 5} + \\frac{2}{s - 1} - \\frac{1}{{\\left(s - 1\\right)}^{2}}</script></html>"
      ],
      "text/plain": [
       "1/(s^2 + 4*s + 5) + 2/(s - 1) - 1/(s - 1)^2"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fpf=fpal.partial_fraction(s);fpf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\frac{1}{s^{2} + 4 \\, s + 5}, \\frac{2}{s - 1}, -\\frac{1}{{\\left(s - 1\\right)}^{2}}\\right]</script></html>"
      ],
      "text/plain": [
       "[1/(s^2 + 4*s + 5), 2/(s - 1), -1/(s - 1)^2]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fpf.operands()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Jetzt versuchen wir jeder davon zu transformieren:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[e^{\\left(-2 \\, t\\right)} \\sin\\left(t\\right), 2 \\, e^{t}, -t e^{t}\\right]</script></html>"
      ],
      "text/plain": [
       "[e^(-2*t)*sin(t), 2*e^t, -t*e^t]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "inverses=[]\n",
    "for op in fpf.operands():\n",
    "    op=inverse_laplace(op,s,t)\n",
    "    inverses.append(op)\n",
    "inverses\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-t e^{t} + e^{\\left(-2 \\, t\\right)} \\sin\\left(t\\right) + 2 \\, e^{t}</script></html>"
      ],
      "text/plain": [
       "-t*e^t + e^(-2*t)*sin(t) + 2*e^t"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fpfi=0;\n",
    "for op in inverses:\n",
    "    fpfi=fpfi+op;\n",
    "fpfi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Das Inverse von die originelle Funktion:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-t e^{t} + e^{\\left(-2 \\, t\\right)} \\sin\\left(t\\right) + 2 \\, e^{t}</script></html>"
      ],
      "text/plain": [
       "-t*e^t + e^(-2*t)*sin(t) + 2*e^t"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "inverse_laplace(fpal,s,t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}</script></html>"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bool(inverse_laplace(fpal,s,t)==fpfi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6.3 2. Ordung Anfangswertproblem "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "4y''-4y'+5y=8e^{-t}\\\\\n",
    "y(0)=13\\\\\n",
    "y'(0)-2\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x=var('x')\n",
    "t=var('t')\n",
    "y=function('y')(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-8 \\, e^{\\left(-t\\right)} + 5 \\, y\\left(t\\right) - 4 \\, \\frac{\\partial}{\\partial t}y\\left(t\\right) + 4 \\, \\frac{\\partial^{2}}{(\\partial t)^{2}}y\\left(t\\right)</script></html>"
      ],
      "text/plain": [
       "-8*e^(-t) + 5*y(t) - 4*diff(y(t), t) + 4*diff(y(t), t, t)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "de = 4*diff(diff(y,t),t) -4*diff(y,t)+5*y-8*e**(-t)\n",
    "de"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, s^{2} \\mathcal{L}\\left(y\\left(t\\right), t, s\\right) - 4 \\, s \\mathcal{L}\\left(y\\left(t\\right), t, s\\right) - 4 \\, s y\\left(0\\right) - \\frac{8}{s + 1} + 5 \\, \\mathcal{L}\\left(y\\left(t\\right), t, s\\right) + 4 \\, y\\left(0\\right) - 4 \\, \\mathrm{D}_{0}\\left(y\\right)\\left(0\\right)</script></html>"
      ],
      "text/plain": [
       "4*s^2*laplace(y(t), t, s) - 4*s*laplace(y(t), t, s) - 4*s*y(0) - 8/(s + 1) + 5*laplace(y(t), t, s) + 4*y(0) - 4*D[0](y)(0)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dl=laplace(de,t,s);\n",
    "pdl=dl;\n",
    "pdl"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir kennen dass\n",
    "$$\n",
    "y(0)=13\\\\\n",
    "y'(0)-2\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/lib/python2.7/site-packages/IPython/core/interactiveshell.py:2882: DeprecationWarning: Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future release of Sage; you can use named arguments instead, like EXPR(x=..., y=...)\n",
      "See http://trac.sagemath.org/5930 for details.\n",
      "  exec(code_obj, self.user_global_ns, self.user_ns)\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, s^{2} \\mathcal{L}\\left(y\\left(t\\right), t, s\\right) - 4 \\, s \\mathcal{L}\\left(y\\left(t\\right), t, s\\right) - 52 \\, s - \\frac{8}{s + 1} + 5 \\, \\mathcal{L}\\left(y\\left(t\\right), t, s\\right) - 4 \\, \\mathrm{D}_{0}\\left(y\\right)\\left(0\\right) + 52</script></html>"
      ],
      "text/plain": [
       "4*s^2*laplace(y(t), t, s) - 4*s*laplace(y(t), t, s) - 52*s - 8/(s + 1) + 5*laplace(y(t), t, s) - 4*D[0](y)(0) + 52"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pdl=pdl.substitute(y(0)==13);\n",
    "pdl\n",
    "#print(pdl)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, s^{2} \\mathcal{L}\\left(y\\left(t\\right), t, s\\right) - 4 \\, s \\mathcal{L}\\left(y\\left(t\\right), t, s\\right) - 52 \\, s - \\frac{8}{s + 1} + 5 \\, \\mathcal{L}\\left(y\\left(t\\right), t, s\\right) + 60</script></html>"
      ],
      "text/plain": [
       "4*s^2*laplace(y(t), t, s) - 4*s*laplace(y(t), t, s) - 52*s - 8/(s + 1) + 5*laplace(y(t), t, s) + 60"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pdl=4*s^2*laplace(y(t), t, s) - 4*s*laplace(y(t), t, s) - 52*s - 8/(s + 1) + 5*laplace(y(t), t, s) - 4*(-2) + 52 \n",
    "pdl"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lösen mit $\\mathcal{L}(y)$ als Unbekannte:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathcal{L}\\left(y\\left(t\\right), t, s\\right) = \\frac{4 \\, {\\left(13 \\, s^{2} - 2 \\, s - 13\\right)}}{4 \\, s^{3} + s + 5}</script></html>"
      ],
      "text/plain": [
       "laplace(y(t), t, s) == 4*(13*s^2 - 2*s - 13)/(4*s^3 + s + 5)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sdl=solve(pdl,laplace(y(t),t,s));\n",
    "(sdl[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{4 \\, {\\left(161 \\, s - 179\\right)}}{13 \\, {\\left(4 \\, s^{2} - 4 \\, s + 5\\right)}} + \\frac{8}{13 \\, {\\left(s + 1\\right)}}</script></html>"
      ],
      "text/plain": [
       "4/13*(161*s - 179)/(4*s^2 - 4*s + 5) + 8/13/(s + 1)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "psdl=sdl[0].right().partial_fraction(s)\n",
    "psdl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\frac{1}{26} \\, {\\left(322 \\, \\cos\\left(t\\right) - 197 \\, \\sin\\left(t\\right)\\right)} e^{\\left(\\frac{1}{2} \\, t\\right)}, \\frac{8}{13} \\, e^{\\left(-t\\right)}\\right]</script></html>"
      ],
      "text/plain": [
       "[1/26*(322*cos(t) - 197*sin(t))*e^(1/2*t), 8/13*e^(-t)]"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "il=[]\n",
    "for o in psdl.operands():\n",
    "    il.append(inverse_laplace(o,s,t))\n",
    "il    \n",
    "#inverse_laplace(sdl[0],s,t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{26} \\, {\\left(322 \\, \\cos\\left(t\\right) - 197 \\, \\sin\\left(t\\right)\\right)} e^{\\left(\\frac{1}{2} \\, t\\right)} + \\frac{8}{13} \\, e^{\\left(-t\\right)}</script></html>"
      ],
      "text/plain": [
       "1/26*(322*cos(t) - 197*sin(t))*e^(1/2*t) + 8/13*e^(-t)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ill=0;\n",
    "for i in il:\n",
    "    ill+=i\n",
    "ill"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Überprüfung:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(K_{2} \\cos\\left(t\\right) + K_{1} \\sin\\left(t\\right)\\right)} e^{\\left(\\frac{1}{2} \\, t\\right)} + \\frac{8}{13} \\, e^{\\left(-t\\right)}</script></html>"
      ],
      "text/plain": [
       "(_K2*cos(t) + _K1*sin(t))*e^(1/2*t) + 8/13*e^(-t)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "desolve(de,[y,t])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We substitute our result inside the function manually, with t=0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-2</script></html>"
      ],
      "text/plain": [
       "-2"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(ill,t).subs(t==0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}13</script></html>"
      ],
      "text/plain": [
       "13"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ill.subs(t==0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}8 \\, e^{\\left(-t\\right)} - 8 \\, e^{t}</script></html>"
      ],
      "text/plain": [
       "8*e^(-t) - 8*e^t"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u=(4*diff(diff(ill,t),t)-4*diff(ill,t)+5*ill-8*e**t)\n",
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}0</script></html>"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u.subs(t==0)"
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "SageMath 8.1",
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   "name": "sagemath"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
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   "file_extension": ".py",
   "mimetype": "text/x-python",
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