Slope+fields

__ Slope Fields __ Do you want to solve and understand those pesky differential equations? Do they ever confuse and hinder you from developing your mathematical education? Never fear! Slope Fields are here! media type="youtube" key="r7VA_4JRPR4" height="385" width="480"

So, in order to make a slope field, you must start with a differential equation, like in the video. From there, you plug in coordinate values into the respective x and y values of the differential equation. After that is said and done, you get a number value, which is the slope of the tangent line at that particular point. Graph a little linear line with that slope on the coordinate point and repeat for as many coordinates as needed to get the general slope field of a differential equation. For Example: The differential equation:
 * X Values || Y Values || [[image:http://www.sitmo.com/gg/latex/latex2png.2.php?z=100&eq=%5Cfrac%7Bdy%7D%7Bdx%7D]] ||  ||   ||
 * 0 || 0 || 0 ||  ||   ||
 * 1 || 0 || 1 ||  ||   ||
 * 2 || 0 || 2 ||  ||   ||
 * 0 || 1 || 1 ||  ||   ||
 * 0 || 2 || 2 ||  ||   ||
 * 1 || 1 || 2 ||  ||   ||
 * 1 || 2 || 3 ||  ||   ||
 * 2 || 1 || 3 ||  ||   ||
 * 2 || 2 || 4 ||  ||   ||
 * -1 || 1 || 0 ||  ||   ||
 * ||  ||   || You can calculate all of the other values, then graph the slopes you get to come up with a slope field like the one below. ||   ||

This is the a slope field of the differential equation,



As you see slope fields can have different graphs depending on where the graph starts. If you start anywhere on the top half, you get a concave up parabola, where as if you start on the bottom, the graph will be concave down. Also, as you can see, the bigger x is, the wider the parabola is. So depending on which point on the slope field you start on, you can have a whole new graph entirely!

So usually, a differential equation gives a certain condition such as then it gives the differential, After that, graph of the slope field using prior mentioned methods, then, take the initial condition, and plug it into the slope field. draw the graph from the point.
 * [[image:http://www.vlab.com/autog/newsletter/newsletter_images/ns2_image/3-5.gif link="http://www.vlab.com/autog/newsletter/newsletter_images/ns2_image/3-5.gif"]] ||

That would be the third circle out. That is how you use slope fields to graph a differential equation.