autonomous versus non-autonomous equation

Some rules of change depend on WHAT you have but not on WHEN it is; others depend on the clock too. A radioactive sample decays at a rate set purely by how much is left — midnight or noon makes no difference. A solar panel's output, by contrast, depends explicitly on the time of day. The first kind of equation is autonomous, the second non-autonomous, and the distinction shapes how their solutions behave.

Precisely, a first-order equation is autonomous if the independent variable does not appear explicitly on the right-hand side: it has the form y' = f(y), the rate depends on the state y alone. It is non-autonomous if the independent variable shows up explicitly, y' = f(t, y). So y' = k y is autonomous (no t on the right), while y' = k y + sin(t) is non-autonomous (the forcing term sin(t) brings the clock in). The word 'autonomous' means 'self-governing': the system's law is fixed for all time, unchanging as the clock ticks.

Autonomy buys real simplicity. Because the rule is time-independent, an autonomous equation's slope field looks the same in every vertical strip — shift any solution in time and you get another solution — so the entire long-term story collapses onto a single line, the phase line, organized by the equilibria. This is why autonomous equations are where stability analysis lives. Non-autonomous equations, with their explicit time dependence, generally lack this shift symmetry and need the fuller machinery; recognizing which type you face tells you which qualitative tools are available.