Consider the following PDE "system" consisting of a single pde
Two generalized integrating factors are
Note that is already the divergence of a function, so that a constant (the number 1 in the result above) is an integrating factor. To verify for correctness these integrating factors use
The conserved currents are related to the generalized integrating factors via Sigma mu[alpha, n] pde[n] = Divergence J[alpha] = 0. These are the J[alpha] corresponding to the mu[alpha] computed above; they depend on arbitrary functions
To verify these results use
An example where the integrating factor depends on an arbitrary function
For this example, integrating factors up to order 1, that is, depending at most on first order derivatives, are
which is in agreement with the general result obtained first. This is a related conserved current of order 1
Specifying directly the functionality expected also confirms that there is no non-trivial integrating factor depending only on and but there is one depending on an arbitrary function of , and
In various cases it is simpler, or of more use, to compute integrating factors of polynomial type, or with a mathematical function dependency on the field of functions of the input system. For these purposes use the option typeofintegratingfactor = ... where the right-hand-side can be polynomial or functionfield. For example, for , a polynomial integrating factor, presented without specializing the arbitrary constants (option split = false) is
The following application of Euler's operator to shows that is already a divergence of a function
This is a conserved current with the same functionality of the last integrating factor computed and a verification of the result