differential equation question

Question

For a phrase line to be a meaningful tool, explain why it is essential for the differential equation to be autonomous

Answer 1

Intuitively, an autonomous equation allows you to determine the different possible tangent slopes of the graph of a solution passing through any point in the xy-plane because y' is a function of x and y in an autonomous differential equation. This quite directly allows one to observe the phase lines associated with the ODE by following the direction indicated by the induced grid of tangent slopes. One could not do this if the ODE is not an autonomous one.