A heuristic method for solving linear and quasi-linear first order partial differential equations

We present a heuristic method for finding first integrals for linear and quasi-linear first order partial differential equations, when we are often lead to solve exact ordinary differential equations. The method is inspired by the corresponding one used for integrating first order exact ordinary differential equations of the form

and is applied for integrating first order exact ordinary differential equations in the case of three variables:

To our best knowledge, the method is new at least from the point of view of its theoretical and applicative features presented in full details in the following.