{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "8504f001-72c2-4b36-9a0f-a24ff35bd093",
   "metadata": {},
   "source": [
    "# Lineare Differentialgleichungen"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "efb9194d-6c66-4fa6-8164-490334248ac2",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-danger\">\n",
    "<b>Ziel des Notebooks:</b>\n",
    "\n",
    "- Grundlegende Konzepte der gewöhnlichen linearen Differentialgleichungen (DGL) verstehen\n",
    "    - Definition\n",
    "    - Lösungen einer DGL\n",
    "        - explizit von DGL 1. Ordnung\n",
    "    - Struktur der Lösungsräume\n",
    "- Rechnen mit linearen DGL \n",
    "    - Werkzeuge erarbeiten, um lineare Systeme von DGL zu lösen\n",
    "        - Jordan-Normalform\n",
    "        - Matrixexponential\n",
    "\n",
    "<br>\n",
    "    \n",
    "<b>Mathematische Tools:<b>\n",
    "- Analysis I & II\n",
    "- Lineare Algebra I & II\n",
    "\n",
    "    \n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3db6bca2-f23f-49fe-924a-386cff75eeae",
   "metadata": {},
   "source": [
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "272c784d-1cba-4794-9e86-dd2f2d49766b",
   "metadata": {},
   "source": [
    "Haben Sie sich jemals gefragt, wie man alltägliche Prozesse mathematisch modellieren kann? Z.B. wie bewegt sich ein einfaches Pendel? Wie verändert sich eine Hasen -und Fuchspopulation mit der Zeit? Wie verläuft die Bahn der Erde um die Sonne?\n",
    "\n",
    "Falls Sie sich dafür interessieren, dann sind u.a. lineare Differentialgleichungen (DGL) Ihre Antwort auf diese Frage. All diese Prozesse lassen sich approximativ über lineare Differentialgleichungen modellieren. \n",
    "\n",
    "Historisch gesehen hatte man erstmals Differentialgleichungen genutzt, um bestimmte Bewegungstrajektorien eines Objektes besser zu beschreiben. Schon Isaac Newton betrachtete früher Bewegungen mit Reibung, die von der Geschwindigkeit (also der Ableitung einer Funktion des Ortes) abhängt, und war so gezwungen, mit Differentialgleichungen zu rechnen.\n",
    "Erst Augustin Louis Cauchy konnte im 19. Jahrhundert durch seine Arbeiten mit Grenzwerten, Ableitungen und Integralen ein Fundament für die Theorie der Differentialgleichungen liefern.\n",
    "In der Praxis ist man sehr an Differentialgleichungen interessiert, da viele Prozesse mit diesen modelliert werden können.\n",
    "\n",
    "***\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "308aeb5a-4a53-4aa7-9459-61d915b35919",
   "metadata": {},
   "source": [
    "Wir beginnen zunächst mit einem einfachen Beispiel eines schwingenden Pendels. \n",
    "\n",
    "<br>\n",
    "\n",
    "Wir bringen eine Kugel der Masse $m$ mithilfe eines gedanklich masselosen Fadens an ein festes Fundament. Danach heben wir die Kugel an und lassen sie wieder los. Sie können sich natürlich schon denken, was passieren wird: Die Kugel schwingt hin und her. Aber können Sie aus den gegebenen Startbedingungen (z.B. Höhe der Kugel, Länge des Fadens) schon bestimmen wie schnell die Schwingung stattfindet? Wir werden später sehen, dass wir durch Differentialgleichungen sehr viel Information gewinnen können.\n",
    "\n",
    "Um den Vorgang besser zu verstehen, schieben wir kurz das 2. Newtonsche Gesetz aus der klassischen Physik ein. Das Gesetz besagt, dass in einem gewissen Bezugssystem (für die Interessierten: Inertialsystem) jeder Körper nach der Gleichung\n",
    "\n",
    "$$ \\vec F = m \\cdot \\ddot{\\vec x}$$\n",
    "\n",
    "handelt. Dabei steht $\\vec F$ für alle von außen wirkenden Kräfte (welche wären das in unserem Fall?), $m$ für die Masse und $\\vec x \\in \\mathbb R^3$ für den Ortsvektor des betrachteten Objekts. Die zwei Punkte über dem Vektor benutzt man in der Physik häufig, um die zweite Ableitung nach der Zeit abzukürzen. \n",
    "\n",
    "Um die Kräfte ausfindig zu machen, betrachten wir die untenstehende Skizze.\n",
    "\n"
   ]
  },
  {
   "attachments": {
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"
    }
   },
   "cell_type": "markdown",
   "id": "8c40a877-4221-4097-a96a-ec4da1707ba8",
   "metadata": {},
   "source": [
    "\n",
    "<div>\n",
    "    <img src=\"attachment:bf4e7364-4036-4466-a34f-c2eec8672568.png\" width=\"700\"/>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "68a64353-6f6a-4b0e-a937-7195055ed833",
   "metadata": {},
   "source": [
    "Es wirkt auf die Kugel die Gewichtskraft $\\vec F_G  =  m \\cdot \\begin{pmatrix} 0 \\\\ 0 \\\\ -g \\end{pmatrix}$, wobei $g \\approx 9,81 \\frac{m}{s^2}$ (das ist die Beschleunigung!) entspricht und die dritte Komponente den Anteil entlang $\\vec e_3$ beschreibt.\n",
    "Weiter hält der Faden die Kugel fest und wirkt der Radialkraft $F_{rad}$ mit der Seilkraft $ F_{Seil}$ entgegen. Die resultierende Kraft äußert sich wie in der Skizze eingezeichnet mit $F_{tan} = -mg \\cdot \\sin(\\varphi(t))$. Dabei ist $\\varphi(t)$ eine Funktion der Zeit, die einen Winkel zurückgibt. Beispielsweise $\\varphi(3) = 3 $ rad. \n",
    "\n",
    "Weil der Faden mit der Kugel verbunden ist, zwingt sie die Kugel auf eine kreisförmige Bahn, wobei sich der Radius nicht ändert, da wir annehmen, dass der Faden seine Länge nicht ändert. Wir können daher die Trajektorie der Kugel schon alleine durch den Winkel beschreiben: $\\ddot{x}_{tan} = l \\cdot \\ddot\\varphi(t)$, wobei $l$ die Länge des Fadens ist. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f7194e2-e0d0-4c53-b05f-8e849084f7c8",
   "metadata": {},
   "source": [
    "Insgesamt erhälten wir also \n",
    "\n",
    "\\begin{align*}\n",
    "F &= m\\ddot x \\\\\n",
    "F_{tan} &= m\\cdot \\ddot \\varphi(t) \\\\\n",
    "-mg \\cdot sin(\\varphi(t)) &= ml\\cdot \\ddot\\varphi(t) \n",
    "\\end{align*}\n",
    "\n",
    "und folglich eine Gleichung der Form $$\\ddot \\varphi(t) + \\frac{g}{l} \\cdot \\sin(\\varphi(t)) = 0.$$\n",
    "\n",
    "Das ist eine Gleichung unserer gesuchten Funktion $\\varphi$, die auch die Ableitung von $\\varphi$ enthält. Eine solche Gleichung nennt man **Differentialgleichung**.\n",
    "\n",
    "Lösen wir diese Gleichung, d.h. finden wir eine Funktion $\\varphi$, die diese Bedingung erfüllt, so können wir aus den Informationen von $\\varphi$ bereits die Bahnkurve beschreiben! "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fc698e23-b7ab-4375-bea4-4ae62f427c06",
   "metadata": {},
   "source": [
    "Wir wollen eine solche Gleichung erst einmal sauber definieren, bevor wir anfangen sie tatsächlich zu lösen. Schließlich gibt es einige Dinge, die man vorher wissen sollte. Dabei beschränken wir uns hier auf lineare gewöhnliche Differentialgleichungen. Es gibt noch eine Vielzahl anderer Arten von DGL, die jedoch den Rahmen des Notebooks sprengen würden. \n",
    "\n",
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "936cf016-6ccd-4b60-8982-19a30b05c627",
   "metadata": {},
   "source": [
    "Man bezeichnet eine DGL als *gewöhnlich*, falls die gesuchte Funktion nur von **einer** Variablen abhängt. Die DGL ist *linear*, wenn die Gleichung Funktionen und deren Ableitungen **linear** enthält - d.h. $y$ sowie die Ableitungen kommen nur in 1. Potenz in einem Summanden vor. Die folgende Definition fasst dies kurz zusammen."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "38d80fa2-c7eb-4b2a-988a-573f11f2f31c",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "\n",
    "**Lineare Gewöhnliche Differentialgleichungen** sind in der Form \n",
    "\n",
    "$$ y^{(n)}(x) = \\sum^{n-1}_{k=0} a_k(x) y^{(k)}(x) + g(x),$$\n",
    "\n",
    "wobei $y$ eine auf einem offenen Intervall $I \\subset \\mathbb R$ definierte Funktion ist und $y^{(n)}$ für die n-te Ableitung steht. Meist sind die $y$ reell-, komplex- oder vektorwertig und die $a_k$ reell- oder komplexwertig. D.h. $y \\in C(I, \\mathbb R^m)$ oder $y \\in C(I, \\mathbb C^m)$ und $m\\in \\mathbb N$. Falls $g \\equiv 0$ gilt, so nennt man die DGL homogen, andernfalls inhomogen. </div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c6bcd25e-6025-44a3-8c52-e60510187df3",
   "metadata": {},
   "source": [
    "Häufig lässt man der Übersicht wegen die Variable einfach weg. Aus dem Kontext der gewöhnlichen DGL wird klar, dass die Funktion sowieso nur von einer Variable abhängt. Wie wir später sehen werden, ist die Wahl des Intervalls $I$ für die Eindeutigkeit und Existenz der Lösungen wichtig.\n",
    "\n",
    "Beispielsweise schreibt man $$ y'' = 42y' + 69y + 7x$$ statt $$y''(x) = 42y'(x) + 69y(x) +7x.$$\n",
    "\n",
    "Zudem fällt Ihnen wahrscheinlich auf, dass die $a_k(x)$ konstant sind. Wir werden uns der Einfachheit halber zunächst mit konstanten Koeffizienten befassen."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "01dfc7ca-f996-4e0f-bf0b-5de8e85d439b",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "### Aufgabe\n",
    "\n",
    "***\n",
    "\n",
    "Sind folgende Gleichungen lineare gewöhnliche Differentialgleichungen?\n",
    "\n",
    "1. $y^{(42)}+exp(x)y = sin(\\frac{1}{x})$\n",
    "2. $y'y''+ y = 0$\n",
    "3. $y'^2+ y' - y = 0$\n",
    "\n",
    "\n",
    "\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "52810aad-d412-442b-8f66-12a1ef5aaac7",
   "metadata": {},
   "source": [
    "Bevor wir uns mit den Techniken zur Lösung der Differentialgleichungen beschäftigen, wollen wir die Struktur dieser DGL ein wenig näher untersuchen und verstehen, welche Eigenschaften die Lösungen besitzen.\n",
    "Dazu möchten wir ein kleines Beispiel betrachten. Es soll\n",
    "\n",
    "$$y' - y = 0$$\n",
    "\n",
    "gelöst werden. Man sieht vielleicht schnell ein, dass für $y_1(x) = \\exp(x)$ die Gleichung erfüllt ist. **Allerdings** ist das nicht die einzige Lösung. Beispielsweise wäre $y_2(x) = C \\cdot \\exp(x)$ mit $C \\in \\mathbb R$ genauso gut eine Lösung. Wir sehen sogar noch mehr: Setzt man $y_1+y_2$ in die Gleichung ein, so wird diese ebenfalls gelöst! Man könnte also vermuten, dass der Raum der Lösungen dieser DGL einen Vektorraum bildet."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5037d514-c894-4808-9861-fe366ff414b2",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-success\">\n",
    "    \n",
    "### Aufgabe\n",
    "\n",
    "***\n",
    "\n",
    "Zeigen Sie, dass für $y\\in C(I,\\mathbb R)$ der Lösungsraum der Gleichung $$y^{(n)}(x) = \\sum^{n-1}_{k=0} a_k(x) y^{(k)}(x)$$ einen $\\mathbb R$-Vektorraum bildet. \n",
    "\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "50104bc5-8996-4710-af9c-0282af9c0ae2",
   "metadata": {},
   "source": [
    "Analog zeigt man diese Aussage auch für $y\\in C(I, \\mathbb C^m)$ etc.\n",
    "Da wir jetzt wissen, dass der Lösungsraum der homogenen DGL einen Vektorraum bildet, kann man sich die Frage nach einer Basis stellen. Eine solche Basis für den Lösungsraum nennt man **Fundamentalsystem**. Hat man diese Basis mal bestimmt, so steht uns der Lösung unseres Problems nichts mehr im Wege.\n",
    "\n",
    "Um die Differentialgleichungen besser begreifen zu können, setzen wir uns nun mit Differentialgleichungen erster Ordnung auseinander. Denn mit einem kleinen Trick kann man zukünftig lineare Differentialgleichungen höherer Ordnung immer in ein System von DGL erster Ordnung transformieren. So reicht es dann, Differentialgleichungen erster Ordnung zu verstehen. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bdf5cdc5-ffab-4da2-8946-889f2f0affa3",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "\n",
    "\n",
    "Ein **lineares System von DGL erster Ordnung** ist in der Form \n",
    "\n",
    "$$ y_1'(x) = a_{11}(x)y_1(x) + \\dots + a_{1n}(x)y_n(x) + b_1(x),$$\n",
    "$$ y_2'(x) = a_{21}(x)y_1(x) + \\dots + a_{2n}(x)y_n(x) + b_2(x),$$\n",
    "$$\\vdots$$\n",
    "$$ y_n'(x) = a_{n1}(x)y_1(x) + \\dots + a_{nn}(x)y_n(x) + b_n(x).$$\n",
    "\n",
    "    \n",
    "Oder in Vektorform $$\\vec y' = A(x) \\vec y(x) + \\vec b(x)\\quad \\text{mit} \\quad \\vec y = \\begin{pmatrix} y_1 \\\\ \\vdots \\\\y_n \\end{pmatrix},  \\quad \\vec b = \\begin{pmatrix} b_1 \\\\ \\vdots \\\\b_n \\end{pmatrix}.$$\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cc3f81b3-011a-46b7-9c7f-1b0cf0f7c3ef",
   "metadata": {},
   "source": [
    "Wir betrachten nun sogenannte Anfangswertprobleme reellwertiger Funktionen erster Ordnung."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5de2768f-d59b-4f4e-86ac-0c4d2127b066",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "    \n",
    "    \n",
    "Seien $I\\subseteq \\mathbb R, x_0 \\in I, a,g: I \\rightarrow \\mathbb R $ stetig. \n",
    "$$\n",
    "\\begin{align} y'(x) &=  a(x)y(x) + g(x), \\quad x\\in I \\\\\n",
    "y(x_0) &= y_0 \\qquad \\text{(Anfangsbedingung)}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Man nennt ein solches Gleichungssystem mit Anfangsbedingung **Anfangswertproblem**.\n",
    "\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3da0c02f-3ed4-4108-ba67-9802f6061cd0",
   "metadata": {},
   "source": [
    "Zunächst schauen wir uns das homogene Problem an, also $g \\equiv 0$. Dann löst\n",
    "\n",
    "$$ y(x) = y_0 \\exp\\left(\\int_{x_0}^x a(s)ds\\right) = y_0 \\exp(A(x))$$ mit $$ A(x):=  \\int_{x_0}^x a(s)ds$$\n",
    "\n",
    "die homogene Differentialgleichung. Sie können sich durch Einsetzen von $y$ gerne selbst überzeugen. \n",
    "\n",
    "Wie Sie bereits wissen, bildet der Lösungsraum $L$ für die homogene Differentialgleichung einen Vektorraum. Wir haben durch die vorher genannte Lösung eine Basis bestimmt. Um nun die inhomogene Differentialgleichung zu lösen, müssen wir Informationen über $g(x)$ einbauen. Dazu bedient man sich zum Beispiel mit der [Variation der Konstanten](https://en.wikipedia.org/wiki/Variation_of_parameters). "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a2bcd5fb-9e55-4f8b-a9f3-ee1b88f093de",
   "metadata": {},
   "source": [
    "Wir skizzieren hier kurz die Idee an:\n",
    "Zunächst setzt man $y_p(x) = c(x) \\cdot \\exp(A(x))$ an. Der wesentliche Unterschied zur vorherigen Lösung ist, dass wir eine Variation der \"Konstanten\" $c$ erlauben. In dem Sinne ist $c$ also als Funktion zu betrachten.\n",
    "\n",
    "Differenzieren wir $y_p$ so ergibt sich mit der Produktregel\n",
    "\n",
    "$$ y_p(x)' = c(x)' \\cdot \\exp(A(x)) + c(x) \\cdot a(x)\\exp(A(x)) = c(x)' \\cdot \\exp(A(x)) + a(x)y_p(x)$$\n",
    "\n",
    "Die DGL wird also genau dann gelöst, wenn $$ c(x)' \\cdot \\exp(A(x)) = g(x)$$\n",
    "\n",
    "Durch Integrieren können wir das $c(x)$ bestimmen und wir erhalten die Lösung für das inhomogene Problem \n",
    "\n",
    "$$ y(x) = \\exp\\left(\\int_{x_0}^x a(s)ds\\right) \\cdot \\left(\\int_{x_0}^x \\exp\\left(-\\int_s^x a(t)\\,dt \\right) g(s)ds + k \\right)$$\n",
    "\n",
    "oder\n",
    "\n",
    "$$y(x) = e^{A(x)} \\cdot \\left(\\int_{x_0}^x e^{-A(x)} g(s) \\;ds + k \\right)$$\n",
    "\n",
    "wobei $k\\in \\mathbb R$.\n",
    "\n",
    "Die Exponentialfunktion möchten wir uns im Kopf behalten. Sie wird bei Lösungen von lineaeren Systemen von DGL eine zentrale Rolle spielen."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "61e0a6b9-2b6d-47bc-9be8-741d8865aaa3",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "### Aufgabe\n",
    "\n",
    "***\n",
    "\n",
    "Lösen Sie folgende Differentialgleichung.\n",
    "$$\n",
    "y' + xy = x \\quad y(0) = 1\n",
    "$$\n",
    "\n",
    "\n",
    "\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bce86fc9-e96b-44d6-9819-ef9f4ae94baa",
   "metadata": {},
   "source": [
    "Gleichungen müssen nicht unbedingt Lösungen haben. Das gilt auch hier bei den DGL. An dieser Stelle geben wir einen wichtigen Satz mit, der bei gewissen Bedingungen die Existenz und Eindeutigkeit einer Lösung garantiert.\n",
    "Man kann damit einschätzen, wie man an einer DGL vorgehen muss und ob Vorsicht geboten ist. Man möchte schließlich nicht mühsam nach einer Lösung suchen, die vielleicht gar nicht existiert."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2f948511-7783-4da6-a42a-ee168d83dd25",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "\n",
    "**Satz (Picard-Lindelöf)**\n",
    "\n",
    "Sei $D \\subseteq \\mathbb R \\times \\mathbb R^n$ abgeschlossen mit $(x_0,y_0) \\in int(D)$  (das [Innere](https://de.wikipedia.org/wiki/Innerer_Punkt) von $D$) und $f: D \\rightarrow \\mathbb R^n$ stetig in $x$ differenzierbar und in $y$ Lipschitz-stetig. Dann existiert ein $\\varepsilon>0$, sodass \n",
    "    \n",
    "$$\n",
    "y'(x) = f(x, y(x)) \\quad y(x_0) = y_0\n",
    "$$\n",
    "    \n",
    "im Intervall $[x_0-\\varepsilon,x_0+\\varepsilon]$ eine **eindeutige** Lösung besitzt.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3cac058e-2d6b-4b85-991e-bf45b304a3e9",
   "metadata": {},
   "source": [
    "Anfangswertprobleme besitzen unter gewissen Voraussetzungen immer eine Lösung. Falls $D$ nicht angegeben wird, so versteht man des Öfteren $D=\\mathbb R \\times \\mathbb R^n$. Man ist generell daran interessiert eine globale Lösung ausfindig zu machen.\n",
    "Dass die Eindeutigkeit einer Lösung nicht immer gewährleistet ist, wollen wir uns nun mit einem Beispiel vergewissern."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c026a673-c1c3-44fe-b515-3d410a7fd478",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "### Aufgabe\n",
    "\n",
    "***\n",
    "\n",
    "Wir betrachten das AWP\n",
    "\n",
    "$$\n",
    "y' = ay^{\\frac{2}{3}} \\quad y(0)=0.\n",
    "$$\n",
    "\n",
    "\n",
    "Zeigen Sie, dass $y(t) = 0$ und $$ y(t)=\n",
    "    \\begin{cases}\n",
    "        \\left(\\frac{at}{3}\\right)^3 & \\text{if } t <0 \\\\\n",
    "        \\;\\;\\,0 & \\text{if } x \\geq 0\n",
    "    \\end{cases}\n",
    "$$\n",
    "    \n",
    "die DGL lösen. Warum kann man mit Picard-Lindelöf nicht zeigen, dass das AWP eine lokal eindeutige Lösung besitzt?\n",
    "    \n",
    "Für welche Anfangsbedingungen $(t_0,y_0)$, also $y(t_0) = y_0$, existiert eine lokale Lösung? \n",
    "    \n",
    "***    \n",
    "    \n",
    "Zeigen Sie mit Picard-Lindelöf, dass \n",
    "    \n",
    "$$\n",
    "y' = y^2+4y-1 \\quad y(0) = -3 \\in \\mathbb R\n",
    "$$\n",
    "\n",
    "lokal eine eindeutige Lösung besitzt.\n",
    "\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "64b1e05d-00fd-45ca-9a3c-b04705739bcf",
   "metadata": {},
   "source": [
    "Nun kennen wir ein Kriterium, wann Differentialgleichungen eine Lösung besitzen können. Wir sind nun speziell an diesen Lösungen interessiert und setzen den Fokus für dieses Notebook auf Systeme linearer homogener Differentialgleichungen mit konstanten Koeffizienten.\n",
    "\n",
    "Erinnern wir uns nochmal an eine solche Differentialgleichung\n",
    "\n",
    "$$\\vec y' = A \\vec y(x) \\quad \\text{mit} \\quad \\vec y = \\begin{pmatrix} y_1 \\\\ \\vdots \\\\y_n \\end{pmatrix}.$$\n",
    "\n",
    "\n",
    "Sie kommt der linearen Differentialgleichung 1. Ordnung  \n",
    "\n",
    "$$y' = ay$$ \n",
    "\n",
    "ziemlich ähnlich. Sie haben vorhin gesehen, dass die Exponentialfunktion eine solche homogene Gleichung löst. Kann man das auf unser System an DGL übertragen?\n",
    "Glücklicherweise lautet die Antwort darauf ja!\n",
    "\n",
    "Für die DGL 1. Ordnung ist die Lösung $ y(x) = C\\cdot e^{ax}$ mit $C\\in \\mathbb R$ und wir können unser System an DGL ähnlich lösen: Für ein Intervall $D\\subseteq \\mathbb R$ und $y:D \\rightarrow \\mathbb R^n$ besitzt die vektorwertige DGL die Lösung\n",
    "\n",
    "$$\n",
    "    y(t) = y_0 \\cdot\\exp(At) \\quad y(0) = y_0 \\in \\mathbb R^n.\n",
    "$$\n",
    "\n",
    "Ihnen fällt wahrscheinlich der Term $\\exp(At)$ auf. Wie kann man eine Matrix exponentieren?\n",
    "Wir führen daher das sogenannte Matrixexponential ein:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "893eddc6-1d17-4f7c-b8ce-954c346d9539",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "    \n",
    "\n",
    "Sei A eine reelle -oder komplexe Matrix $n\\times n$-Matrix. Das Matrixexponential ist definiert mit\n",
    "    \n",
    "$$\\exp(A) = \\sum_{k=0}^{\\infty} \\frac{A^k}{k!} = \\mathbb 1_{n\\times n} + A + \\frac{A^2}{2} + \\dots.$$\n",
    "    \n",
    "\n",
    "\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "81bfd1b6-a528-4231-bbbe-116114ae3e1d",
   "metadata": {},
   "source": [
    "Es stellt sich heraus, dass es äußerst schwierig ist, große Potenzen von Matrizen auszurechnen. Jedoch gibt es Formen, in der Matrizen dann doch einfach zu berechnen sind. Dazu ziehen wir die Jordan-Normalform heran.\n",
    "\n",
    "Sei $J \\in \\mathbb R^{nxn}$ die Jordan-Normalform zu $A\\in \\mathbb R^{nxn}$ mit \n",
    "\n",
    "$$\n",
    "A = SJS^{-1}.\n",
    "$$\n",
    "\n",
    "Dann lässt sich das Problem der Berechnung von $A^k$ auf $J^k$ reduzieren."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0f3c5db3-a359-47a8-b425-421c90aedfb1",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "### Aufgabe\n",
    "\n",
    "***\n",
    "Erklären Sie die vorherige Aussage.\n",
    "    \n",
    "Beweisen Sie zusätzlich, dass, falls für $A,B \\in \\mathbb R^{nxn}$ $A\\cdot B = B\\cdot A$ ist, \n",
    "\n",
    "$$\n",
    "    \\exp(A+B) = \\exp(A)\\exp(B)\n",
    "$$\n",
    "\n",
    "gilt.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2656d023-00ad-44d1-8a65-f52c4697d3d9",
   "metadata": {},
   "source": [
    "Die obige Aussage ist sehr nützlich, wenn man das Matrixexponential der Jordanmatrix bestimmen möchte.\n",
    "Betrachten wir beispielsweise \n",
    "\n",
    "$J = \\begin{pmatrix} 2&1&0 \\\\ 0&2&1 \\\\ 0&0&2 \\end{pmatrix} = \\begin{pmatrix} 2&0&0 \\\\ 0&2&0 \\\\ 0&0&2 \\end{pmatrix} + \\begin{pmatrix} 0&1&0 \\\\ 0&0&1 \\\\ 0&0&0 \\end{pmatrix} = D + N.$ \n",
    "\n",
    "Weil $D$ und $N$ kommutieren, gilt also $\\exp(J) = \\exp(D+N) = \\exp(D)\\exp(N)$. Zusätzlich wissen wir, dass $N$ nilpotent ist. Damit ist die Berechnung von $\\exp(N)$ von viel einfacher und für Diagonalmatrizen sowieso.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "98ccffbc-ea08-412a-a2dc-46813a973401",
   "metadata": {},
   "source": [
    "\n",
    "Der einfachste Fall ist also, falls $J$ diagonal ist. Wir schauen uns nun an, wie man ein einfaches System an Differentialgleichungen löst. \n",
    "\n",
    "\n",
    "**Beispiel**\n",
    "\n",
    "Gegeben sei das System\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\dot x(t) &= y(t) - z(t) \\\\\n",
    "\\dot y(t) &= y(t) \\\\\n",
    "\\dot z(t) &= x(t) -y(t) +3z(t)\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "mit $x(0)=y(0)=z(0)=1$.\n",
    "\n",
    "Wir können das System umschreiben und erhalten\n",
    "\n",
    "$$\n",
    "   \\begin{pmatrix} \\dot x \\\\ \\dot y\\\\ \\dot z \\end{pmatrix} = \\begin{pmatrix} 0 & 1 & -1 \\\\ 0&1&0 \\\\ 1&-1&3 \\end{pmatrix} \\cdot \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} =: A\\cdot \\vec y.\n",
    "$$\n",
    "\n",
    "\n",
    "Nun müssen wir das Matrixexponential $\\exp(At)$ berechnen. Dazu bilden wir die Jordan-Normalform von $A$ und erhalten $ J= \\begin{pmatrix} 1&0&0 \\\\ 0&1&0 \\\\ 0&0&2 \\end{pmatrix}$ sowie $S= \\begin{pmatrix} 1 & 0 & 1 \\\\ 1&2&0 \\\\ 0&1&-1 \\end{pmatrix}$. Daraus folgt nun\n",
    "\n",
    "$$\n",
    "\\exp(At) = \\sum_{k=0}^{\\infty} \\frac{A^k \\cdot t^k}{k!} = S^{-1}\\cdot \\sum_{k=0}^{\\infty} \\frac{J^k \\cdot t^k}{k!}\\cdot S = S^{-1} \\begin{pmatrix} \\exp(t)&0&0 \\\\ 0&\\exp(t)&0 \\\\ 0&0&\\exp(2t) \\end{pmatrix} S = \\begin{pmatrix} \\exp(t)&\\exp(2t)&\\exp(t)+\\exp(2t) \\\\ 3 \\exp(t)&4 \\exp(t)& \\exp(t) \\\\ \\exp(t) & 2 \\exp(t) + \\exp(2t) & \\exp(2t) \\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "\n",
    "Also ist unsere Lösung \n",
    "\n",
    "$$\n",
    "    \\vec y =  \\begin{pmatrix} x(t) \\\\  y(t) \\\\  z(t) \\end{pmatrix} = \\exp(At)\\cdot y_0  =    \\begin{pmatrix} \\exp(t)&\\exp(2t)&\\exp(t)+\\exp(2t) \\\\ 3 \\exp(t)&4 \\exp(t)& \\exp(t) \\\\ \\exp(t) & 2 \\exp(t) + \\exp(2t) & \\exp(2t) \\end{pmatrix} \\begin{pmatrix} 1 \\\\ 1 \\\\ 1\\end{pmatrix} = \\begin{pmatrix} 2 \\exp(t) + 2 \\exp(2t) \\\\  8 \\exp(t) \\\\  3 \\exp(t) + 2 \\exp(2t) \\end{pmatrix}.\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "22b3e158-7272-46eb-ab93-c2b227d0b4bf",
   "metadata": {},
   "source": [
    "Wir können damit jede einzelne Lösung komponentenweise ablesen. \n",
    "\n",
    "Wie lösen wir aber eine Differentialgleichung der Form \n",
    "$\n",
    "    y''' + y'' + y' + y = 0 \\text{?}\n",
    "$\n",
    "\n",
    "Wir definieren dazu neue Variablen und schreiben die Differentialgleichung so um, dass man ein System an Differentialgleichungen erhält.\n",
    "\n",
    "Setze also $w_0=y$, $w_1=y'$, $w_2=y''$. Dann ist folgende Gleichung äquivalent zu unserer ursprünglichen Differentialgleichung\n",
    "\n",
    "$$\n",
    "\\begin{pmatrix}  w_0' \\\\ w_1' \\\\ w_2' \\end{pmatrix} = \\begin{pmatrix} 0&1&0 \\\\ 0&0&1 \\\\ -1&-1&-1 \\end{pmatrix} \\cdot \\begin{pmatrix}  w_0 \\\\ w_1 \\\\ w_2 \\end{pmatrix}.\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9b97b2ad-ed63-4d41-8b26-915ff8689026",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-success\">\n",
    "\n",
    "### Aufgabe\n",
    " ***\n",
    "Verallgemeinern Sie das Verfahren für eine DGL der Form \n",
    "    \n",
    "$$\n",
    "\\sum_{k=0}^{N} a_k y^{(k)} = 0.\n",
    "$$\n",
    "    \n",
    "Wie sieht die zugehörige Matrix aus? \n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "355ac5d5-d2f4-4777-9845-b3c1af4601b3",
   "metadata": {},
   "source": [
    "***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc021d96-56d5-4af8-9a90-0cbdc8d28180",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-danger\">\n",
    "    \n",
    "### Rückblick-Aufgabe\n",
    "***\n",
    "\n",
    "Das Ziel des Notebooks war, zu verstehen, was Lineare Differentialgleichungen sind und welche Eigenschaften sie besitzen. Wir haben gelernt, dass man jede Differentialgleichung höherer Ordnung in ein System von DGL 1.Ordnung umschreiben kann. In unserem Spezialfall mit konstanten Koeffizienten haben wir solche Systeme von DGL mithilfe der Jordan-Normalform und des Matrixexponentials gelöst. \n",
    "    \n",
    "\n",
    "Gehen Sie nochmal zurück zum Anfang des Notebooks und betrachten Sie die Differentialgleichung. Jetzt können Sie mit Ihren Kenntnissen diese Gleichung (mit einem kleinen Zusatz eines Reibungsterms) lösen:\n",
    "    \n",
    "$$\n",
    "\\ddot \\varphi(t) + \\frac{g}{l} \\cdot \\sin(\\varphi(t)) = k\\cdot \\dot\\varphi(t)\n",
    "$$\n",
    "    \n",
    "Nehmen Sie dazu an, dass für kleine Winkel $\\sin(\\varphi(t)) \\approx \\varphi(t)$ ist.\n",
    "Was sagen Ihnen die Ergebnisse? Wie wird sich unser Pendel also bewegen?\n",
    "</div>"
   ]
  },
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   "source": [
    "***"
   ]
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    "### Literaturhinweise\n",
    "- Wikipedia (2023). *Lineare gewöhnliche Differentialgleichung*. https://de.wikipedia.org/wiki/Lineare_gewöhnliche_Differentialgleichung\n",
    "\n",
    "- Wikipedia (2023). *Fundamentalsystem (Mathematik).* https://de.wikipedia.org/wiki/Fundamentalsystem_(Mathematik)\n",
    "\n",
    "- Wikipedia (2023). *Harmonischer Oszillator.* https://de.wikipedia.org/wiki/Harmonischer_Oszillator\n"
   ]
  }
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