Why are autonomous differential equations important?

Question

A very soft question:

A separable first order differential equation is a differential equation that in the form $y'=f(y)g(x)$, with $f$ and $g$ continuous. They are important, since we have an efficient way to solve them on any interval where $f(y)$ is not $0$. The price to pay is the computation of two antiderivatives. Many examples of equations coming from physics are separable.

Autonomous equations are the special case $y'=f(y)$. As a measure of the importance people may give to this notion, many textbooks will include this terminology and devote some space to it. And Wikipedia has a specific article on the topic.

My question: **why do we care about autonomous equations (sufficiently to give them a specific name)? **

Sure there are examples of autonomous equations in physics and other sciences. But the general separable case is simple enough (in the sense that if someone is able to understand the autonomous case, it seems difficult to understand why he would have troubles with the general situation). Why bother with this apparently superfluous extra-terminology?

Note : When writing this question, I was assuming functions to be real-valued. If one needs to extend the context to complex-valued or vector-valued, or to systems of equations for an answer, please feel free to do it.

Answer 1

To some extent, you are being misled by just looking at a single differential equation rather than a system of them. Most of the time, there's more than one quantity changing and you need a system to describe it. And an autonomous system $$ \eqalign{\frac{dx_1}{dt} &= f_1(x_1, x_2, \ldots, x_n)\cr \frac{dx_2}{dt} &= f_2(x_1,x_2, \ldots, x_n)\cr \ldots & \ldots\cr \frac{dx_n}{dt} &= f_n(x_1,x_2,\ldots,x_n)} $$ is quite different from a separable equation.

Autonomous differential equations (or systems of them) describe a system whose behaviour does not change with time. Thus what the system does now with a given initial value is the same as what it would do tomorrow with the same initial value. This is a common enough situation that it is worth investigating.

Autonomous equations and systems also have some very useful properties: even when you can't find formulas for the solutions (and, contrary to what you tend to see in introductory courses, most realistic differential equation models don't have such formulas), you can often say a lot about their behaviour in the long run by looking at equilibrium solutions and their stability, phase plane analysis etc.