This article deals with the method to get the general solutions to homogeneous Euler equations of order 2, then generalizes the results to the case of order n.

The treatment,by which the non-homogeneous equation was transformed into homogeneous equation,not only simplifies.

Considering the symplectic relations of variations in the thermal equilibrium formulations and gradient equations,the non-homogeneous Hamilton canonical equation was transformed into homogeneous equation for solving independently the coupling problem of piezothermoelastic bodies by increasing the dimensions of the canonical equation.

First of all,a non-linear Schrodinger equation can be converted into homogeneous equations,and then the precise integration method can be used to solve these problems.

The Differentiator Series Solution to Euler Equation;

The simplified constant-transform method for sloving Euler equation of order three;

The governing equation is 3D Euler equations,which is solved by the finite volume method.