In this paper,we discuss and obtain an analytic intuitionistic proof method of the variables chang formula in double integrals and give a simple proof method.

We get a new q-series identity by using Bailey′s transformation formula and Ismail′s result.

The transformation formulas of a Gaussian beam through para- bolic diverging GRIN rod lens are derived by using the variationl method and ABCD law.

By using the deformation theory,the transformation formulas for the Bergman kernels on real analytic mappings are researched,and the weighted Bergman kernel on a new domain can be obtained from a known Bergman kernel by using the transformation formulas for the Bergman kernels.

With variable substitution,several kinds of linear differential equations are solved and several sufficient and necessary conditions are obtained.

The method searching for solutions to first order non liner equation with variable substitution and integral gene, and the method searching for solutions to all the integral with integral gene were discussed and expounded in this paper.

This paper uses variable substitution to discuss the methods of solving three integral kinds of function-coefficient differential equations of the third order.