{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Lecture 4\n", "The new python in this class is just drawing solution curves. Consider the ODE: \n", "$$\n", "\\frac{dx}{dt} \n", "= \\frac{3t^2 + 4t + 2}{2(t-1)}.\n", "$$\n", "We found that it has solution given implicitly by: \n", "$$\n", "x^2 - 2x - t^3 - 2t^2 - 2t = C. \n", "$$\n", "\n", "For a fixed value of $C$, this equation defines a curve in the $(t,x)$ plane. We would like to use python to plot some\n", "sample curves. \n", "\n", "The first step is to make a meshgrid for a range of t and x values. I got the ones I did below by \n", "experimenting. Below, T and X are both matrices and so Z is a matrix of the same dimension. The \n", "entries of Z are just $x^2 - 2x - t^3 - 2t^2 - 2t$ where $t$ and $x$ come from the corresponding entries \n", "in the T and X matrices. \n", "\n", "The ax.contour(T,X,Z [3]) function looks at all of the values in the Z matrix. If these values are equal to \n", "3, then it draws a dot at the corresponding point from the T and Z matrix. Then it connects those dots \n", "(actually, if draws the dot if the value of Z is \"close\" to 3). The [3] is an array that contains only \n", "3 and it says to draw the level set when the \"$C$\" is $3$.\n", "\n", "(Notice if you change the .2 to 2 you will get \n", "a much less smooth curve). " ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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