Linear_ODE_real_evs.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "# Use pretty printing for SymPy `expr`essions\n",
    "init_printing()\n",
    "\n",
    "def noop(*args, **kwargs):\n",
    "    pass\n",
    "if __name__ != '__main__':\n",
    "    display = noop"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear ODEs: example with different real eigenvalues\n",
    "\n",
    "In this lecture we solve the linear ODE $\\dot x=Ax$, where $A=\\begin{bmatrix}1&2\\\\2&1\\end{bmatrix}$.\n",
    "I will solve this ODE manually verifying each step using [Python] library [SymPy].\n",
    "\n",
    "[Python]: https://www.python.org \"Python programming language\"\n",
    "[SymPy]: https://www.sympy.org \"Symboic math in Python\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Eigenvalues and eigenvectors\n",
    "\n",
    "Its [determinant] equals $-3$ and its [trace] equals $2$, hence its [characteristic polynomial] is $\\lambda^2-2\\lambda-3$. The [eigenvalues] of $A$ are the roots of this polynomial: $3$ and $-1$.\n",
    "\n",
    "[determinant]: https://en.wikipedia.org/wiki/Determinant\n",
    "[trace]: https://en.wikipedia.org/wiki/Trace_(linear_algebra)\n",
    "[characteristic polynomial]: https://en.wikipedia.org/wiki/Characteristic_polynomial\n",
    "[eigenvalues]: https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 2\\\\2 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  2⎤\n",
       "⎢    ⎥\n",
       "⎣2  1⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOCAYAAAD9lDaoAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3UlEQVQoFW2RvRGCQBCFD7UAxhIgNXI0J8AO1BK0BEJItQMMTTU0w4DE1A6gBMYOzu+dg4PIzrzZ27dvf+7Os9aaLMsCY0wCZAvQgAT+KWIMJMghtlEU3cCpLMsZ3Bn/IK5HBAewB1+jQF1f4CJSohhUJHwRHbtz9uEDiRTUBKocMt9o8SGkaVoBUtaN+6um6xxSF3I31rgh08JXxEeX7I9iRA6KLv/TicodlVP8qtt+0gYk1pxD/KbDaa/P4iS06BLffk2rU2HjMV/qAui9+hZTGGqcBBJqn765D34Dzrdx4/BA7EwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 2$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAOCAYAAADaOrdAAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+klEQVQ4Ea2U2w2CQBBFwViAtEAJRjuwBGkBSjB+wS8laAvagdiB2Ip0gOeSgawmJMbdSSbz2Hguszsx7vs+8rWqqlIYhXFWRNU1/Ua92FcEkKACjiIR+Z7eBc/IrwsSX8sB5AYeWcMEFEc1Qog84XTmYmoS1ZMtp+zPBKC+OnF/7kx1Uj/EJC5fU+xo1HhBftah9ySCyACuCRLY4rrCBz6Y93aNIDci+LFdcVmWWsE7rviraTX1tbPG+YtDMRPvSeyaom9R6hsCur4sxMO3gFqgszcR4uE7RBpEFF3bWNGEEDm4ZOUI6uE1mda4834Tgw53r9wsJU5/kG9LAlkZChbCfAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle -3$"
      ],
      "text/plain": [
       "-3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\operatorname{PurePoly}{\\left( \\lambda^{2} - 2 \\lambda - 3, \\lambda, domain=\\mathbb{Z} \\right)}$"
      ],
      "text/plain": [
       "PurePoly(lambda**2 - 2*lambda - 3, lambda, domain='ZZ')"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ -1, \\  3\\right]$"
      ],
      "text/plain": [
       "[-1, 3]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = Matrix([[1, 2], [2, 1]])\n",
    "display(A, A.trace(), A.det(), A.charpoly(), solve(A.charpoly().as_expr()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solving $Ax=\\lambda x$ for $\\lambda=-1, 3$, we find the corresponding eigenvectors $\\begin{bmatrix}-1\\\\1\\end{bmatrix}$ and $\\begin{bmatrix}1\\\\1\\end{bmatrix}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left( -1, \\  \\left\\{ x_{1} : - x_{2}\\right\\}\\right), \\  \\left( 3, \\  \\left\\{ x_{1} : x_{2}\\right\\}\\right)\\right]$"
      ],
      "text/plain": [
       "[(-1, {x₁: -x₂}), (3, {x₁: x₂})]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('x1 x2')\n",
    "x = Matrix([x1, x2])\n",
    "[(λ, solve(A * x - λ * x)) for λ in sorted(A.eigenvals())]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course, `Python` can find eigenvalues and eigenvectors of a matrix in one line"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ -1 : 1, \\  3 : 1\\right\\}$"
      ],
      "text/plain": [
       "{-1: 1, 3: 1}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left( -1, \\  1, \\  \\left[ \\left[\\begin{matrix}-1\\\\1\\end{matrix}\\right]\\right]\\right), \\  \\left( 3, \\  1, \\  \\left[ \\left[\\begin{matrix}1\\\\1\\end{matrix}\\right]\\right]\\right)\\right]$"
      ],
      "text/plain": [
       "⎡⎛       ⎡⎡-1⎤⎤⎞  ⎛      ⎡⎡1⎤⎤⎞⎤\n",
       "⎢⎜-1, 1, ⎢⎢  ⎥⎥⎟, ⎜3, 1, ⎢⎢ ⎥⎥⎟⎥\n",
       "⎣⎝       ⎣⎣1 ⎦⎦⎠  ⎝      ⎣⎣1⎦⎦⎠⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(A.eigenvals(), A.eigenvects())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Diagonalization\n",
    "\n",
    "Therefore, $A=PDP^{-1}$, where $P=\\begin{bmatrix}-1&1\\\\1&1\\end{bmatrix}$ and $D=\\begin{bmatrix}-1&0\\\\0&3\\end{bmatrix}$. Here the columns of $P$ are the eigenvectors of $A$ and $D$ is the diagonal matrix with the eigenvalues of $A$ at the diagonal. Again, we can ask [Python] to find $P$ and $D$ without asking it for the eigenvectors first. See also Wikipedia page about [matrix diagonalization](https://en.wikipedia.org/wiki/Diagonalizable_matrix).\n",
    "\n",
    "[Python]: https://www.python.org \"Python programming language\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-1 & 1\\\\1 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-1  1⎤\n",
       "⎢     ⎥\n",
       "⎣1   1⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-1 & 0\\\\0 & 3\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-1  0⎤\n",
       "⎢     ⎥\n",
       "⎣0   3⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "P, D = A.diagonalize()\n",
    "display(P, D)\n",
    "assert P * D * P ** -1 == A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Formula for the solution\n",
    "\n",
    "Recall that solutions of $\\dot y=Dy$ have the form $y(t)=M(t)y(0)$, where $M(t)=\\begin{bmatrix}e^{-t}&0\\\\0&e^{3t}\\end{bmatrix}$.\n",
    "For each solution $y(t)$ of $\\dot y=Dy$, the function $x(t)=Py(t)=PM(t)y(0)$ is a solution of $\\dot x=Ax$.\n",
    "Note that $x(0)=PM(0)y(0)=Py(0)$, hence $y(0)=P^{-1}x(0)$.\n",
    "Finally,\n",
    "$$\n",
    "x(t)=PM(t)P^{-1}x(0)=\\frac{1}{2}\\begin{bmatrix}e^{3t}+e^{-t}&e^{3t}-e^{-t}\\\\e^{3t}-e^{-t}&e^{3t}+e^{-t}\\end{bmatrix}x(0).\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}e^{- t} & 0\\\\0 & e^{3 t}\\end{matrix}\\right], \\  \\left[\\begin{matrix}c_{0}\\\\c_{1}\\end{matrix}\\right], \\  \\left[\\begin{matrix}c_{0} e^{- t}\\\\c_{1} e^{3 t}\\end{matrix}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛⎡ -t      ⎤        ⎡    -t ⎤⎞\n",
       "⎜⎢ℯ     0  ⎥  ⎡c₀⎤  ⎢c₀⋅ℯ   ⎥⎟\n",
       "⎜⎢         ⎥, ⎢  ⎥, ⎢       ⎥⎟\n",
       "⎜⎢      3⋅t⎥  ⎣c₁⎦  ⎢    3⋅t⎥⎟\n",
       "⎝⎣ 0   ℯ   ⎦        ⎣c₁⋅ℯ   ⎦⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{e^{3 t}}{2} + \\frac{e^{- t}}{2} & \\frac{e^{3 t}}{2} - \\frac{e^{- t}}{2}\\\\\\frac{e^{3 t}}{2} - \\frac{e^{- t}}{2} & \\frac{e^{3 t}}{2} + \\frac{e^{- t}}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ 3⋅t    -t   3⋅t    -t⎤\n",
       "⎢ℯ      ℯ    ℯ      ℯ  ⎥\n",
       "⎢──── + ───  ──── - ───⎥\n",
       "⎢ 2      2    2      2 ⎥\n",
       "⎢                      ⎥\n",
       "⎢ 3⋅t    -t   3⋅t    -t⎥\n",
       "⎢ℯ      ℯ    ℯ      ℯ  ⎥\n",
       "⎢──── - ───  ──── + ───⎥\n",
       "⎣ 2      2    2      2 ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}c_{0} \\left(\\frac{e^{3 t}}{2} + \\frac{e^{- t}}{2}\\right) + c_{1} \\left(\\frac{e^{3 t}}{2} - \\frac{e^{- t}}{2}\\right)\\\\c_{0} \\left(\\frac{e^{3 t}}{2} - \\frac{e^{- t}}{2}\\right) + c_{1} \\left(\\frac{e^{3 t}}{2} + \\frac{e^{- t}}{2}\\right)\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡   ⎛ 3⋅t    -t⎞      ⎛ 3⋅t    -t⎞⎤\n",
       "⎢   ⎜ℯ      ℯ  ⎟      ⎜ℯ      ℯ  ⎟⎥\n",
       "⎢c₀⋅⎜──── + ───⎟ + c₁⋅⎜──── - ───⎟⎥\n",
       "⎢   ⎝ 2      2 ⎠      ⎝ 2      2 ⎠⎥\n",
       "⎢                                 ⎥\n",
       "⎢   ⎛ 3⋅t    -t⎞      ⎛ 3⋅t    -t⎞⎥\n",
       "⎢   ⎜ℯ      ℯ  ⎟      ⎜ℯ      ℯ  ⎟⎥\n",
       "⎢c₀⋅⎜──── - ───⎟ + c₁⋅⎜──── + ───⎟⎥\n",
       "⎣   ⎝ 2      2 ⎠      ⎝ 2      2 ⎠⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}c_{0} \\left(\\frac{3 e^{3 t}}{2} - \\frac{e^{- t}}{2}\\right) + c_{1} \\left(\\frac{3 e^{3 t}}{2} + \\frac{e^{- t}}{2}\\right)\\\\c_{0} \\left(\\frac{3 e^{3 t}}{2} + \\frac{e^{- t}}{2}\\right) + c_{1} \\left(\\frac{3 e^{3 t}}{2} - \\frac{e^{- t}}{2}\\right)\\end{matrix}\\right], \\  \\left[\\begin{matrix}2 c_{0} \\left(\\frac{e^{3 t}}{2} - \\frac{e^{- t}}{2}\\right) + c_{0} \\left(\\frac{e^{3 t}}{2} + \\frac{e^{- t}}{2}\\right) + c_{1} \\left(\\frac{e^{3 t}}{2} - \\frac{e^{- t}}{2}\\right) + 2 c_{1} \\left(\\frac{e^{3 t}}{2} + \\frac{e^{- t}}{2}\\right)\\\\c_{0} \\left(\\frac{e^{3 t}}{2} - \\frac{e^{- t}}{2}\\right) + 2 c_{0} \\left(\\frac{e^{3 t}}{2} + \\frac{e^{- t}}{2}\\right) + 2 c_{1} \\left(\\frac{e^{3 t}}{2} - \\frac{e^{- t}}{2}\\right) + c_{1} \\left(\\frac{e^{3 t}}{2} + \\frac{e^{- t}}{2}\\right)\\end{matrix}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛⎡   ⎛   3⋅t    -t⎞      ⎛   3⋅t    -t⎞⎤  ⎡     ⎛ 3⋅t    -t⎞      ⎛ 3⋅t    -t⎞\n",
       "⎜⎢   ⎜3⋅ℯ      ℯ  ⎟      ⎜3⋅ℯ      ℯ  ⎟⎥  ⎢     ⎜ℯ      ℯ  ⎟      ⎜ℯ      ℯ  ⎟\n",
       "⎜⎢c₀⋅⎜────── - ───⎟ + c₁⋅⎜────── + ───⎟⎥  ⎢2⋅c₀⋅⎜──── - ───⎟ + c₀⋅⎜──── + ───⎟\n",
       "⎜⎢   ⎝  2       2 ⎠      ⎝  2       2 ⎠⎥  ⎢     ⎝ 2      2 ⎠      ⎝ 2      2 ⎠\n",
       "⎜⎢                                     ⎥, ⎢                                   \n",
       "⎜⎢   ⎛   3⋅t    -t⎞      ⎛   3⋅t    -t⎞⎥  ⎢   ⎛ 3⋅t    -t⎞        ⎛ 3⋅t    -t⎞\n",
       "⎜⎢   ⎜3⋅ℯ      ℯ  ⎟      ⎜3⋅ℯ      ℯ  ⎟⎥  ⎢   ⎜ℯ      ℯ  ⎟        ⎜ℯ      ℯ  ⎟\n",
       "⎜⎢c₀⋅⎜────── + ───⎟ + c₁⋅⎜────── - ───⎟⎥  ⎢c₀⋅⎜──── - ───⎟ + 2⋅c₀⋅⎜──── + ───⎟\n",
       "⎝⎣   ⎝  2       2 ⎠      ⎝  2       2 ⎠⎦  ⎣   ⎝ 2      2 ⎠        ⎝ 2      2 ⎠\n",
       "\n",
       "      ⎛ 3⋅t    -t⎞        ⎛ 3⋅t    -t⎞⎤⎞\n",
       "      ⎜ℯ      ℯ  ⎟        ⎜ℯ      ℯ  ⎟⎥⎟\n",
       " + c₁⋅⎜──── - ───⎟ + 2⋅c₁⋅⎜──── + ───⎟⎥⎟\n",
       "      ⎝ 2      2 ⎠        ⎝ 2      2 ⎠⎥⎟\n",
       "                                      ⎥⎟\n",
       "        ⎛ 3⋅t    -t⎞      ⎛ 3⋅t    -t⎞⎥⎟\n",
       "        ⎜ℯ      ℯ  ⎟      ⎜ℯ      ℯ  ⎟⎥⎟\n",
       " + 2⋅c₁⋅⎜──── - ───⎟ + c₁⋅⎜──── + ───⎟⎥⎟\n",
       "        ⎝ 2      2 ⎠      ⎝ 2      2 ⎠⎦⎠"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('c0 c1') # Coordinates of $x(0)$\n",
    "var('t') # Time\n",
    "M = Matrix([[exp(D[0,0] * t), 0], [0, exp(D[1,1] * t)]])\n",
    "c = Matrix([c0, c1])\n",
    "y = M * c\n",
    "display((M, c, y))\n",
    "assert y.diff(t) == D * y\n",
    "assert y.subs(t, 0) == c\n",
    "x = P * M * P ** -1 * c\n",
    "display(P * M * P ** -1, # Fundamental matrix of solutions\n",
    "        x) # Solution with `x(0)=c`\n",
    "assert x.diff(t).expand() == (A * x).expand()\n",
    "assert x.subs(t, 0) == c\n",
    "x.diff(t), A * x"
   ]
  }
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