Autonomous vs Non-Autonomous equations

The formulation of the W-method, in its as-presented form, can be used only with equations that do not incorporate the dependent variable in them. Such equations that are functions of just the independent variables are referred to as autonomous functions.

When the equation is a function of both the dependent & independent variables, they are referred to as non-autonomous functions.

An example of a non-autonomous ODE is the famous stiff ODE from Curtiss & Hirschfelder (Integration of stiff equations).

 

Non-autonomous equations can be converted into autonomous ones by defining the independent variable as a dependent one. The above ODE then becomes:

 

The above two equations can then be used with the W-method (or any method that is represented in autonomous form).