This was a review for Wednesday's Hour Exam.
I began by using dy/dx = -y/x as an example. I talked about direction fields, isoclines, and used these to draw some solutions. Then I set about solving the equation.
1. It's separable, and with some care to signs the general solution is y=c/x (or, somewhat better, the "implicit" solution xy=c, since that includes (for c=0) the case of the y-axis).
2. It's also homogeneous: the isoclines are straight lines through 0. The substitution y=sx leads to the same solutions.
3. It's linear. I pointed out that any homogeneous linear equation is separable; the integrating factor method is only useful for solving inhomogeneous equations. In fact the IF satisfies a homogeneous linear equation which differs by a sign from the associated homogeneous equation.
4. It's not autonomous. I wrote down x-dot = x(1-x) (and commented on the x-dot notation) and solved it using partial fractions. Notice that the constant solutions are lost when you divide. One is returned to you when you replace +-e^{t+c} by ce^t (c=0, ie, x=0); the other corresponds to c=\infty.
But then I reminded the class that the main point about autonomous equations was the ease with which you can get qualitative information. I drew the phase line, reminded them about critical points stable and unstable, added "neutrally stable" for the case in which g(x) doesn't change sign at the critical point, and then pointed out that by the chain rule the points of inflection of solutions correspond to critical points of g: in the present example, x=1/2.
Finally I spoke briefly about the sources of ODEs we have encountered.