muchange(tr, Mu, y(x), [t, u(t)])

1. The following nonlinear second order ODE

has the following integrating factor.

This integrating factor can be tested using mutest:

We now change variables

so that the ODE becomes the following.

The integrating factor for this transformed ODE can be obtained from the integrating factor of ODE1 as follows:

2. muchange works as follows. Consider for instance a first order ODE turned exact by means of an integrating factor

and now consider the most general point transformation of variables:

Perform a change of variables in the exact_ODE as a whole:

The left hand side in the above equation will also be an exact ODE if we multiply it by the denominator of the right hand side:

The new integrating factor for the transformed ODE is, generally speaking, given by

It is easy to see that this result is valid irrespective of the differential order of the ODE under consideration.

3. Consider the most general second order ODE having an integrating factor depending on (x, y'); this ODE is given by (see redode)

What will be the integrating factor if we interchange the roles of the dependent and independent variables? The related transformation is given by

The new integrating factor is

Transform the ODE and test the new integrating factor:

(Note that, when testing an integrating factor, mutest tests it against .)

DEtools[Lie], DEtools[mutest], dsolve/Lie, PDEtools[dchange]