Is there a procedure to solve Diophantine Equations?

Question

How would you go about solving a multivariable, non-linear Diophantine Equation?

Answer 1

By a famous result of Matiyasevich, there is no universal algorithm which, when fed any Diophantine equation, will determine whether or not that equation has a solution in integers.

Interestingly, it is still unknown whether there is an algorithm that will always determine whether or not a Diophantine equation has a solution in rationals.

For quadratic Diophantine equations, in any number of variables, there is an algorithm. For degree $4$ equations, it is known that there is not. The question for cubic equations is unresolved.

Answer 2

No, there is no general procedure. This was Hilbert's 10th problem.