Autonomous Differential Equations

A differential equation is called autonomous if it does not explicitly involve the independent variable \( x \), but depends only on the unknown function \( y(x) \) and its derivatives: $$ F(y, y', \dots, y^{(n)}) = 0 $$ In this context, \( y \) denotes \( y(x) \), and the same applies to its derivatives.

Illustrative Example

Consider the differential equation:

$$ y'(x) - y(x) = 0 $$

This is an autonomous equation, as it does not contain the independent variable \( x \) explicitly.

Example 2

Now consider the equation:

$$ y'(x) - y(x) \cdot x = 0 $$

This equation is not autonomous, since the independent variable \( x \) appears explicitly in the right-hand term, multiplying the unknown function \( y(x) \).

Key Observation

If \( y(x) \) is a solution to an autonomous differential equation, then any horizontal translation of that function - i.e., any function of the form \( y(x + c) \), where \( c \in \mathbb{R} \) - is also a solution of the same equation: $$ y(x + c), \quad c \in \mathbb{R} $$

This property reflects the fact that the behavior of the system described by the equation does not depend on the specific value of the independent variable, but only on the state of the system itself.

And so forth.