Linear_ODEs.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear differential equations\n",
    "\n",
    "MAT332, Fall 2020, Yury Kudryashov, Week 2, Lecture 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dimension 1\n",
    "\n",
    "Consider the equation $\\dot x=ax$, where $a$ is a real number and $x$ is an unknown funnction $\\mathbb R\\to\\mathbb R$. It is easy to see that $x(t)=e^{at}$ is a solution of this equation:\n",
    "$$\n",
    "\\frac{d}{dt}e^{at}=e^{at}\\cdot \\frac{d(at)}{dt}=ae^{at}.\n",
    "$$\n",
    "Next, $x(t)=ce^{at}$ is a solution of this equation for any $c$. Are there any other solutions? The answer is NO: for any solution we have $x(t)=x(0)e^{at}$. Indeed, if $x$ is a solution, then\n",
    "$$\n",
    "\\frac{d}{dt}\\left(x(t)e^{-at}\\right)=\\dot x(t)e^{-at}+x(t)\\cdot \\frac{d}{dt}e^{-at}=ax(t)e^{-at}-ax(t)e^{-at}=0,\n",
    "$$\n",
    "hence $x(t)e^{-at}$ is a constant function, $x(t)e^{-at}=x(0)e^{-a\\cdot 0}=x(0)$, thus $x(t)=x(0)e^{at}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also ask `Python` to verify that $x(t)=e^{at}$ is a solution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAz0lEQVQoFXWS4Q2CMBCFhQkIbqAjqBvgBsoGOkf/GUbQFXQENjAyAhtA2AC/V3tGG2hyeXdfH71LSzKO48KWc64KeYeuiQrWiiVmBLyoL+hDG2iGiO3J2zTAM5qZKbAB1UdX1d6IHolGIFpP6kKnm7EA9JFJpZ8PLdIwy4TnD+U6MQ9IM82tb+s5g/GlTpyazQzWrdOM1lL3Fi9jn3tktyZWsYvaTqzteu7A7YRxA2vU1RtJboAePZiZXG1L4iT2+9ba0E8xEPopdoTe3r/YGx/SQ0OZAIYmAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('a t c')\n",
    "x = c * exp(a * t)\n",
    "simplify(x.diff(t) - a * x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Practice problems\n",
    "\n",
    "1. Find all solutions of the equation $\\dot x=2x$.\n",
    "2. Find the solution of $\\dot x=-x$ such that $x(1)=e$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dimension 2\n",
    "\n",
    "Now consider an equation of the form\n",
    "$$\n",
    "\\dot x=Ax,\n",
    "$$\n",
    "where $A$ is a $(2\\times 2)$-matrix and $x\\colon\\mathbb R\\to\\mathbb R^2$, $x(t)=\\begin{pmatrix}x_1(t)\\\\x_2(t)\\end{pmatrix}$ is an unknown vector-valued function.\n",
    "\n",
    "Before discussing the general case, consider a few special cases."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Diagonal matrix $A=\\begin{pmatrix}\\lambda&0\\\\0&\\mu\\end{pmatrix}$\n",
    "\n",
    "In this case the system takes the form\n",
    "$$\n",
    "\\begin{align}\n",
    "\\dot x_1&=\\lambda x_1;\\\\\n",
    "\\dot x_2&=\\mu x_2,\n",
    "\\end{align}\n",
    "$$\n",
    "and we can solve these two equations independently: $x_1(t)=x_1(0)e^{\\lambda t}$, $x_2(t)=x_2(0)e^{\\mu t}$. We can also write the answer as\n",
    "$$\n",
    "x(t)=\\begin{pmatrix}e^{\\lambda t}&0\\\\0&e^{\\mu t}\\end{pmatrix}x(0).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, we can ask `Python` to double check our computation. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}λ & 0\\\\0 & μ\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡λ  0⎤\n",
       "⎢    ⎥\n",
       "⎣0  μ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}e^{t λ} & 0\\\\0 & e^{t μ}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ t⋅λ      ⎤\n",
       "⎢ℯ      0  ⎥\n",
       "⎢          ⎥\n",
       "⎢       t⋅μ⎥\n",
       "⎣ 0    ℯ   ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}c_{0} e^{t λ}\\\\c_{1} e^{t μ}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡    t⋅λ⎤\n",
       "⎢c₀⋅ℯ   ⎥\n",
       "⎢       ⎥\n",
       "⎢    t⋅μ⎥\n",
       "⎣c₁⋅ℯ   ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0⎤\n",
       "⎢ ⎥\n",
       "⎣0⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}c_{0}\\\\c_{1}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡c₀⎤\n",
       "⎢  ⎥\n",
       "⎣c₁⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "var('λ μ t c0 c1')\n",
    "c = Matrix([c0, c1])\n",
    "A = Matrix([[λ, 0], [0, μ]])\n",
    "M = Matrix([[exp(λ * t), 0], [0, exp(μ * t)]])\n",
    "x = M * c\n",
    "display(A) # display matrix `A`\n",
    "display(M) # display the fundamental matrix of solutions\n",
    "display(x) # display the solution `x`\n",
    "display(x.diff(t) - A * x) # verify that `x` is a solution\n",
    "display(x.subs(t, 0)) # verify that `x(0)=(c0, c1)`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Jordan cell $A=\\begin{pmatrix}\\lambda&1\\\\0&\\lambda\\end{pmatrix}$\n",
    "\n",
    "In this case the system takes the form\n",
    "$$\n",
    "\\begin{align}\n",
    "\\dot x_1&=\\lambda x_1+x_2;\\\\\n",
    "\\dot x_2&=\\lambda x_2.\n",
    "\\end{align}\n",
    "$$\n",
    "While these equations are no longer independent, we can solve the second one, then substitute the solution in the first one. Then we get $x_2(t)=x_2(0)e^{\\lambda t}$, $\\dot x_1=\\lambda x_1+x_2(0)e^{\\lambda t}$. One can verify that $x_1(t)=x_2(0)te^{\\lambda t}+x_1(0)e^{\\lambda t}$ is a solution of this differential equation. As we will see later, this is the only solution of this equation with given $x_1(0)$. Again, we can write down the answer as\n",
    "$$\n",
    "x(t)=\\begin{pmatrix}e^{\\lambda t}&te^{\\lambda t}\\\\0&e^{\\lambda t}\\end{pmatrix}x(0)\n",
    "$$\n",
    "and let `Python` verify the answer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}λ & 1\\\\0 & λ\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡λ  1⎤\n",
       "⎢    ⎥\n",
       "⎣0  λ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}e^{t λ} & t e^{t λ}\\\\0 & e^{t λ}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ t⋅λ     t⋅λ⎤\n",
       "⎢ℯ     t⋅ℯ   ⎥\n",
       "⎢            ⎥\n",
       "⎢        t⋅λ ⎥\n",
       "⎣ 0     ℯ    ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}c_{0} e^{t λ} + c_{1} t e^{t λ}\\\\c_{1} e^{t λ}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡    t⋅λ         t⋅λ⎤\n",
       "⎢c₀⋅ℯ    + c₁⋅t⋅ℯ   ⎥\n",
       "⎢                   ⎥\n",
       "⎢          t⋅λ      ⎥\n",
       "⎣      c₁⋅ℯ         ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}c_{0} λ e^{t λ} + c_{1} t λ e^{t λ} - λ \\left(c_{0} e^{t λ} + c_{1} t e^{t λ}\\right)\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡      t⋅λ           t⋅λ     ⎛    t⋅λ         t⋅λ⎞⎤\n",
       "⎢c₀⋅λ⋅ℯ    + c₁⋅t⋅λ⋅ℯ    - λ⋅⎝c₀⋅ℯ    + c₁⋅t⋅ℯ   ⎠⎥\n",
       "⎢                                                 ⎥\n",
       "⎣                        0                        ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0⎤\n",
       "⎢ ⎥\n",
       "⎣0⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}c_{0}\\\\c_{1}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡c₀⎤\n",
       "⎢  ⎥\n",
       "⎣c₁⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = Matrix([[λ, 1], [0, λ]])\n",
    "M = Matrix([[exp(λ * t), t * exp(λ * t)], [0, exp(λ * t)]])\n",
    "x = M * c\n",
    "display(A) # display matrix `A`\n",
    "display(M) # display the fundamental matrix of solutions\n",
    "display(x) # display the solution `x`\n",
    "display(x.diff(t) - A * x) # verify that `x` is a solution\n",
    "display(simplify(x.diff(t) - A * x)) # verify that `x` is a solution\n",
    "display(x.subs(t, 0)) # verify that `x(0)=(c0, c1)`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Complex eigenvalues $A=\\begin{pmatrix}a&-b\\\\b&a\\end{pmatrix}$\n",
    "\n",
    "In this case it is hard to guess the solution (at least without using complex exponentials, see the textbook) but it is easy to verify that\n",
    "$$\n",
    "\\begin{align}\n",
    "x_1(t)&=x_1(0)e^{at}\\cos(bt)-x_2(0)e^{at}\\sin(bt),\\\\\n",
    "x_2(t)&=x_1(0)e^{at}\\sin(bt)+x_2(0)e^{at}\\cos(bt)\n",
    "\\end{align}\n",
    "$$\n",
    "is a solution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}a & - b\\\\b & a\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡a  -b⎤\n",
       "⎢     ⎥\n",
       "⎣b  a ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}e^{a t} \\cos{\\left(b t \\right)} & - e^{a t} \\sin{\\left(b t \\right)}\\\\e^{a t} \\sin{\\left(b t \\right)} & e^{a t} \\cos{\\left(b t \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ a⋅t             a⋅t         ⎤\n",
       "⎢ℯ   ⋅cos(b⋅t)  -ℯ   ⋅sin(b⋅t)⎥\n",
       "⎢                             ⎥\n",
       "⎢ a⋅t            a⋅t          ⎥\n",
       "⎣ℯ   ⋅sin(b⋅t)  ℯ   ⋅cos(b⋅t) ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}c_{0} e^{a t} \\cos{\\left(b t \\right)} - c_{1} e^{a t} \\sin{\\left(b t \\right)}\\\\c_{0} e^{a t} \\sin{\\left(b t \\right)} + c_{1} e^{a t} \\cos{\\left(b t \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡    a⋅t                a⋅t         ⎤\n",
       "⎢c₀⋅ℯ   ⋅cos(b⋅t) - c₁⋅ℯ   ⋅sin(b⋅t)⎥\n",
       "⎢                                   ⎥\n",
       "⎢    a⋅t                a⋅t         ⎥\n",
       "⎣c₀⋅ℯ   ⋅sin(b⋅t) + c₁⋅ℯ   ⋅cos(b⋅t)⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}a c_{0} e^{a t} \\cos{\\left(b t \\right)} - a c_{1} e^{a t} \\sin{\\left(b t \\right)} - a \\left(c_{0} e^{a t} \\cos{\\left(b t \\right)} - c_{1} e^{a t} \\sin{\\left(b t \\right)}\\right) - b c_{0} e^{a t} \\sin{\\left(b t \\right)} - b c_{1} e^{a t} \\cos{\\left(b t \\right)} + b \\left(c_{0} e^{a t} \\sin{\\left(b t \\right)} + c_{1} e^{a t} \\cos{\\left(b t \\right)}\\right)\\\\a c_{0} e^{a t} \\sin{\\left(b t \\right)} + a c_{1} e^{a t} \\cos{\\left(b t \\right)} - a \\left(c_{0} e^{a t} \\sin{\\left(b t \\right)} + c_{1} e^{a t} \\cos{\\left(b t \\right)}\\right) + b c_{0} e^{a t} \\cos{\\left(b t \\right)} - b c_{1} e^{a t} \\sin{\\left(b t \\right)} - b \\left(c_{0} e^{a t} \\cos{\\left(b t \\right)} - c_{1} e^{a t} \\sin{\\left(b t \\right)}\\right)\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡      a⋅t                  a⋅t              ⎛    a⋅t                a⋅t      \n",
       "⎢a⋅c₀⋅ℯ   ⋅cos(b⋅t) - a⋅c₁⋅ℯ   ⋅sin(b⋅t) - a⋅⎝c₀⋅ℯ   ⋅cos(b⋅t) - c₁⋅ℯ   ⋅sin(b\n",
       "⎢                                                                             \n",
       "⎢      a⋅t                  a⋅t              ⎛    a⋅t                a⋅t      \n",
       "⎣a⋅c₀⋅ℯ   ⋅sin(b⋅t) + a⋅c₁⋅ℯ   ⋅cos(b⋅t) - a⋅⎝c₀⋅ℯ   ⋅sin(b⋅t) + c₁⋅ℯ   ⋅cos(b\n",
       "\n",
       "   ⎞         a⋅t                  a⋅t              ⎛    a⋅t                a⋅t\n",
       "⋅t)⎠ - b⋅c₀⋅ℯ   ⋅sin(b⋅t) - b⋅c₁⋅ℯ   ⋅cos(b⋅t) + b⋅⎝c₀⋅ℯ   ⋅sin(b⋅t) + c₁⋅ℯ   \n",
       "                                                                              \n",
       "   ⎞         a⋅t                  a⋅t              ⎛    a⋅t                a⋅t\n",
       "⋅t)⎠ + b⋅c₀⋅ℯ   ⋅cos(b⋅t) - b⋅c₁⋅ℯ   ⋅sin(b⋅t) - b⋅⎝c₀⋅ℯ   ⋅cos(b⋅t) - c₁⋅ℯ   \n",
       "\n",
       "         ⎞⎤\n",
       "⋅cos(b⋅t)⎠⎥\n",
       "          ⎥\n",
       "         ⎞⎥\n",
       "⋅sin(b⋅t)⎠⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0⎤\n",
       "⎢ ⎥\n",
       "⎣0⎦"
      ]
     },
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    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}c_{0}\\\\c_{1}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡c₀⎤\n",
       "⎢  ⎥\n",
       "⎣c₁⎦"
      ]
     },
     "metadata": {},
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    }
   ],
   "source": [
    "var('a b')\n",
    "A = Matrix([[a, -b], [b, a]])\n",
    "M = exp(a * t) * Matrix([[cos(b * t), -sin(b * t)], [sin(b * t), cos(b * t)]])\n",
    "x = M * c\n",
    "display(A) # display matrix `A`\n",
    "display(M) # display the fundamental matrix of solutions\n",
    "display(x) # display the solution `x`\n",
    "display(x.diff(t) - A * x) # verify that `x` is a solution\n",
    "display(simplify(x.diff(t) - A * x)) # verify that `x` is a solution\n",
    "display(x.subs(t, 0)) # verify that `x(0)=(c0, c1)`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Change of basis (a.k.a. conjugate matrix)\n",
    "\n",
    "[Jordan normal form theorem](https://en.wikipedia.org/wiki/Jordan_normal_form) implies that any $(2\\times 2)$-matrix is conjugate to one of the matrices we discussed above. Thus in order to solve $\\dot x=Ax$ with any matrix $A$ it suffices to learn how to deal with each of these cases (done) and how to solve $\\dot x=CAC^{-1}x$ provided that we know the solutions of $\\dot y=Ay$. The answer is simple: for each solution $y(t)$ of $\\dot y=Ay$, the function $x(t)=Cy(t)$ is a solution of $\\dot x=CAC^{-1}x$.\n",
    "\n",
    "Indeed,\n",
    "$$\n",
    "\\frac{d}{dt}(Cy(t))=C\\dot y(t)=CAy(t)=(CAC^{-1})Cy(t).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exponential of a matrix\n",
    "\n",
    "Given an $(n\\times n)$-matrix $A$, one can define $\\exp(A)=e^{A}$ as\n",
    "$$e^{A}=\\sum_{n=0}^{\\infty}\\frac{1}{n!}A^n=I+A+\\frac{A^2}{2!}+\\dots$$,\n",
    "where $n!=1\\times\\dots\\times n$ is the factorial of $n$. Then $x(t)=e^{At}x(0)$ is a solution of $\\dot x=Ax$. Indeed,\n",
    "$$\\frac{d}{dt}(e^{At})=\\frac{d}{dt}\\left(I+tA+\\frac{t^2}{2!}A^2+\\frac{t^3}{3!}A^3+\\dots+\\frac{t^n}{n!}A^n+\\dots\\right)=0+A+tA^2+\\frac{t^2}{2!}A^3+\\dots+\\frac{t^{n-1}}{(n-1)!}A^n+\\dots=Ae^{At},$$\n",
    "hence\n",
    "$$\\frac{d}{dt}(e^{At}x(0))=Ae^{At}x(0).$$\n",
    "\n",
    "Note that computing $e^{At}$ is not a trivial task, so this formula gives us another way to think about the formulas for solutions we discussed above, not a simpler way to compute these solutions."
   ]
  }
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