{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "%matplotlib inline\n", "from pandas import DataFrame\n", "from IPython.display import display,Math,clear_output\n", "from numpy import array,linspace,zeros\n", "from sympy import latex\n", "import matplotlib.pyplot as plt\n", "from IPython.display import HTML" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\n", "
" ] }, "execution_count": 2, "metadata": { }, "output_type": "execute_result" } ], "source": [ "HTML('''\n", "''') " ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### задача 1(118 а)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "$xy\\ d x- (1+x^2) d y=0\\\\\n", "x y\\ d x = (1+x^2) d y \\\\\n", "\\frac{x\\ dx}{1+x^2}=\\frac{dy}{y}\\\\\n", "\\int \\frac{x\\ dx}{1+x^2}=\\int \\frac{dy}{y}\\\\ \n", "производя\\ интегрирование \\\\\n", "\\ \\ \\int \\frac{x\\ dx}{1+x^2}=\\frac{1}{2} \\int \\frac{dx^2}{1+x^2}=\\frac{1}{2} ln(1+x^2)\\\\\n", "\\ \\ \\int \\frac{dy}{y} =ln(y)+C \\\\ \n", "получим \\\\\n", "ln(y)+C=\\frac{1}{2} ln(1+x^2) \\\\ \n", " или\\ собирая\\ логарифмы\\ (где\\ C=-ln(C_{1})): \\\\\n", " ln(y)=ln(C_{1} \\sqrt(1+x^2))\\\\\n", " y=C_{1} \\sqrt(1+x^2)\n", " $" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Задача 2 (118 б)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "$y'= \\frac{2 x}{y}+\\frac{y}{2},\\ y(2)=1,\\ h=0.2\\\\\n", "по\\ формуле:\\\\\n", "y_{i}=y_{i-1}+(x_{i}-x_{i-1})\\ f(x_{i-1},y_{i-1})\\ начиная\\ с\\ x_{0}=2, y_{0}=1\\ получим \\\\ \n", "$" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "\n", " | i=0 | \n", "i=1 | \n", "i=2 | \n", "i=3 | \n", "i=4 | \n", "i=5 | \n", "
---|---|---|---|---|---|---|
$x_{i}$ | \n", "2.0 | \n", "2.2 | \n", "2.400000 | \n", "2.600000 | \n", "2.800000 | \n", "3.000000 | \n", "
$y_{i}$ | \n", "1.0 | \n", "1.9 | \n", "2.553158 | \n", "3.184479 | \n", "3.829511 | \n", "4.504927 | \n", "