I was looking at the following example of reducing a pde
In this case we manage to reduce the order of the pde. There are a couple of more examples that employ this technique of finding the characteristics and parameterising through them toreduce the order of the pde.
My question is, does this always happen? Is this something like completing the square? I can't seem to prove that there will always be reduction in order. If anybody can help me with that I would greatly appreciate it.