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Autonomous Differential Equations: Simple Definition

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What are Autonomous Differential Equations?

Autonomous differential equations have the form

y′ = f(y).

Although the independent variable doesn’t explicitly appear, these equations can also be written as

y′(x) = f(y(x)).

The right side of the equation is independent of x. As there aren’t any terms that depend on x, the equation is self-governing (which is another word for autonomous) [1]. To put that another way, the rate of change of x is only dependent on x itself; It isn’t dependent on time. All autonomous differential equations are characterized by this lack of dependence on the independent variable.

Many systems, like populations, can be modeled by autonomous differential equations. These systems grow and shrink independently—based only on their own behavior and not by any external factors. For example, a population of deer on an island with unlimited resources and no predators is autonomous. However, given a fluctuating number of predators, the system is no longer autonomous because the system now depends on an external factor— the predator population [2].

Properties of Autonomous Systems

All autonomous systems [3]:

  • Model conditions which are constant in time.
  • Are difficult to integrate but easy to solve (for the most part).
  • Are separable. All autonomous equations can be solved (at least implicitly) by separating variables.
  • Contain a wealth of qualitative information even if a solution can’t be found.
  • Are time invariant (a horizontal shift of a solution is another solution). For example, if a solution is y(t) then y(t – t0) is also a solution.

Finding Solutions to Autonomous Differential Equations

Every critical point (also called a stationary point) is a solution of the autonomous differential equation y′ = f(y). A critical point is a real number c so that f(c) = 0.

Complementary functions are one part of the solution to ADE’s.

Watch the following video which shows equilibrium solutions and stability of autonomous systems:

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[1] Autonomous Differential Equations. Retrieved March 20, 2021 from:
[2] Nathan Pflueger (2011). Lecture 27: Autonomous equations. Retrieved March 20, 2021 from:
[3] First Order Autonomous DEs’: Introduction. Retrieved March 20, 2021 from: CC 4.0.

Stephanie Glen. "Autonomous Differential Equations: Simple Definition" From Calculus for the rest of us!

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