The following problem arises in the study of nonlinear mechanics:

The change of variables, , yields the system

However, in many problems the function is simple in form, but only smooth in pieces. If we have an analytical expression for an indefinite integral of the function , that is , then we note that

So we use the change of variables suggested by Lienard,

to get the system

This system has smoother functions than the standard system making it easier to solve numerically.

(See exercise set 3.)