When the right-hand side never mentions the input, the equation says the same thing at every moment: the rate depends on the state alone. The direction field of such an equation is the same along every vertical line, and that one fact has a consequence worth having — sliding a solution sideways gives another solution. The picture is therefore repetitive, and everything worth knowing about it can be read from a single line.

A law of nature with no clock in it: the rule for how fast a population grows depends on how many there are, not on what year it is. Two colonies that start at the same size behave identically, however far apart their starting dates.

The field of $y'=\tfrac{7}{10}(2-y)$ is the same all along every horizontal line, so it repeats sideways. The blue curve is the green one slid to the right, and both are solutions.

An equation is autonomous when it has the form $y'=f(y)$: the rate depends on the state and not on where in the input the state is reached. Every autonomous equation is separable, but what makes them worth a name is that they can be understood without integrating anything, from the sign of $f$ alone.

• The field is constant along horizontal lines, so it repeats sideways, and a shifted solution is a solution: if $y(x)$ solves the equation so does $y(x-a)$ for every $a$.
• A solution is monotone. It can never turn round, because turning would need $f(y)=0$ at the turning value, and that value is itself a constant solution which cannot be crossed.

Autonomous means the right-hand side is a function of the unknown alone. The input may appear on the left, inside the derivative, and nowhere else.

Because the field repeats sideways, a solution moved along the input axis still fits it.

Great work! You now know how to:

• recognise an autonomous equation and say what its field looks like
• use the shift property to make new solutions from old
• argue that solutions of an autonomous equation are monotone
• find the constant solutions and the strips they cut the plane into