section3.ipynb

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    "# Solución lineal de prmer orden \n",
    "\n",
    "una ecuación diferenacial homogenea de primer orden es de la forma \n",
    "\n",
    "$$a(t)y'+b(t)y=c(t)$$\n",
    "si $a(t)\\ne 0$ enonces podemos transformar la ecuación de la forma \n",
    "\n",
    "$$y'(t)+P(t)y(t)=Q(t)$$\n",
    "donde $P(t)=\\frac{b(t)}{a(t)}$ y $Q(t)=\\frac{c(t)}{a(t)}$.\n",
    "\n",
    "Note que si $Q(t)=0$, la ecuación anterior se torna de la forma \n",
    "$$y'(t)+P(t)y(t)=0$$\n",
    "\n",
    "podemos buscar un factor de integración de la forma \n",
    "\n",
    "$$\\rho(t)=e^{\\int P(t)d\\ t},$$\n",
    "note que \n",
    "\n",
    "$$\\frac{d}{dt}\\Big(y(t)e^{\\int P(t)d\\ t}\\Big)=y'e^{\\int P(t)d\\ t}+y(t)P(t)e^{\\int P(t)d\\ t}$$\n",
    "\n",
    "al multiplica a $\\rho(t)$ por la ecuación original tenemos \n",
    "\n",
    "\n",
    "$$y'(t)\\rho(t)+\\rho(t)P(t)y(t)=0$$\n",
    "$$y'(t)e^{\\int P(t)d\\ t}+e^{\\int P(t)d\\ t}P(t)y(t)=0$$\n",
    "$$\\frac{d}{dt}\\Big(y(t)e^{\\int P(t)d\\ t}\\Big)=0$$\n",
    "$$\\int\\frac{d}{dt}\\Big(y(t)e^{\\int P(\\tau)d\\tau}\\Big)d t=C$$\n",
    "de esta forma tenemos que \n",
    "$$y(t)=C\n"
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