The first lecture I listened is about the geometric view of first-degree differential equations and the manual way of creating a direction field.
Function y’=f(x, y) can be shown on the coordinate system in the form of direction field, and one solution of y’=f(x, y), y1(x), shows as one integral curve on the coordinate system. Just as y’=f(x, y) relates to y1(x) (y’ is the derivative of y1(x) and y1(x) is one of many possible integral of y’), the integral curve correspond to y1(x) should relate to the direction field correspond to the differential equation y’=f(x, y); to be specific, the slope of every possible point on the function y1(x) should be the same as the slope shown on that exact point on the direction field. For example, one of the integral of the function y’=-x/y can be y=(4-x2)1/2, shown as a semicircle on the system, and for every point, (0, 2) for example, the slope of that point (which is 0) should be just the same as shown on that point of the direction field of y’=-x/y (meaning there should be a horizontal line on coordinate (0, 2) on the direction field to correspond the semicircle formed by y=(4-x2)1/2. The analytical view of the relationship between y’ and y1 should be just same as the geometric view of that between direction field and integral curve.
Furthermore, the manual way of creating a direction field is also mentioned in the lecture. The key difference between the computer way and human way is the order: the computer picks thousand or tens of thousands of points first, then calculates the slope at every single point and show them on the field. Human, however cannot possibly tolerate such massive amount of calculations. The Manual way start with picking slope, hence the value of y’, then calculate the isocline (a function) where all of its point should have the same slope on the direction field. For example, again y’=-x/y, and we pick the slope y’ of 2, then there goes a function of 2=-x/y and hence y=-x/2, and we draw the line on the coordinate system, then every point on that line should have a slope of 2, which is an extremely efficient way of creating a direction field.