solving for the graceful tree of a graph

The goal of finding a graceful graph is finding numbers to fill in the letters of this graph:

such that the differences between them (to label the edges) are 1,2,3,4,5. Or for larger graphs following that same pattern. 

First one can observe that one equation can be formed from the information as:

But there are five unknowns and that is only one equation, so the way I found to provide the other equations is simply to raise this basic equation to higher powers. So I use:

This provides many solutions but the first one given is:

This solution does work. There might be other ways to find a set of nonlinearly-related equations using the information given that work even better.