Autonomous Equations, Equilibrium Solutions, and Stability
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Autonomous Differential Equations are ones of the form y'=f(y), that is only the dependent variable shows up on the right side. This means the slope fields aka direction fields look the same for all values of the independent variable t. While these are examples of separable differential equations, they allow for a lot of new analysis. We can talk about equilibrium solutions that don't change in time, and we can talk about stability. That is, as time goes to infinity do solution approach or go away from the equilibrium solutions? This is referred to as asymptotic stability.
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