WHAT?!?Originally Posted bydelpin

There is no specific way to solve a Diophantine equation. In fact some Diophantine equations cannot even be proven to they have no solutions! This was one of Hilbert's 23 Problems. Euler was able to prove them/solve them because he was talented.

You might be asking about Differencial equations but just confused the words with Diophantine equations. There is such a thing called Euler's method in the theory of differencial equations, maybe you are asking that?