{
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  {
   "cell_type": "markdown",
   "id": "ceb18a16-df8a-4d3f-8edc-badd6125bdc4",
   "metadata": {},
   "source": [
    "## Euler's method for ODEs"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b547d580-13c0-4eaa-a2f0-2119355d5d7e",
   "metadata": {},
   "source": [
    "Implement the algorithm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4eca1757-9769-44b3-afb5-6b6c1d0a3f64",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "\"\"\"\n",
    "    t, y = myeulers(fun, a, b, n, y1)\n",
    "\n",
    "Solve IVP for ode y' = fun(t, y) on the interrval a <= t <= b, y(a) = y1 \n",
    "\"\"\"\n",
    "function myeulers(fun, a, b, n, y1)\n",
    "    t = range(a, b, n)\n",
    "    y = zeros(n)\n",
    "    h = t[2] - t[1]\n",
    "    y[1] = y1\n",
    "    for i = 1:(n-1)\n",
    "        y[i+1] = y[i] + h * fun(t[i], y[i])\n",
    "    end\n",
    "    return t, y\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9d916dcf-9466-4060-9819-8a98ace31857",
   "metadata": {},
   "source": [
    "Test the code using a differential equation with a known solution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2ef46690-a428-47ba-9d34-8ebf20d13b83",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "a = 0.0\n",
    "b = 5.0\n",
    "n = 128\n",
    "y1 = 0.0\n",
    "fun(t, y) = exp(-sin(t)) - y * cos(t);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "53ae116b-dc6d-4576-968d-cd27bd1662c8",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "exacty(t) = t * exp(-sin(t));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "efbdc4d1-6956-40ef-b53a-c8873dd1e577",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "rest, resy = myeulers(fun, a, b, n, y1);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "49e5464e-5362-465c-bddd-e1fed396981e",
   "metadata": {},
   "source": [
    "Plot the numerical solution and compare it to the exact solution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f982a0b4-c88a-4e56-bc65-6977b1f9d8f4",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "using PyPlot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "bb722b0c-ce7c-4661-92c8-c1bf73c7287b",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "plot(rest, resy, marker=\".\", label=\"Euler's\")\n",
    "plot(rest, exacty.(rest), linestyle=\"dashed\", label=\"Exact\")\n",
    "grid(true)\n",
    "xlabel(\"t\")\n",
    "ylabel(\"y(t)\")\n",
    "legend()\n",
    "title(\"Euler's method for IVP\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9f577a33-5771-4f9d-a65b-faf2d5edee65",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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