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today were going to talk about how to sketch a direction field and then were going to talk about how to sketch a solution through the direction field that passes through a particular point and in this particular problem weve been given the equation y prime or the derivative of Y is equal to Y minus 2x and weve been given the point 1 0 now Ive already taken a sketch of the direction field here because I want to show you guys what it looks like so that you can visualize the final product as we work through the problem but well still work through and tell you how to sketch this basically the first thing you want to do is make sure that your equation is for y Prime in terms of y and X we need an equation for y Prime in order to sketch the direction field because remember that y prime represents the derivative or the slope of our equation Y and the direction field is nothing more than the slope sketched at lots of different points so each of the lines in this direction field represen