Compare and consider kernel estimation that two optimal bandwidths could be chosen and six kernel functions (among them three kernels are often used and famous kernel, other three kernel functions are done by us), which belong to the same function family was used, for half-normal detection function.

Some of optimal bandwidths used depends on detection function, some do not rely on detection function, still use classic ground Maximum Likelihood estimate.

By means of the conditions of total differential equation,in this paper are given the integral factor and general solution of one kind of differential equation and are obtained the differential equations that satisfy the unknown functions in some of total differential equations, thus are found the unknown functions and their general solutions.

In this paper, using the linear substitution of unknown function, obtains a nec-essary and sufficient condition for three order linear homogeneous differential equation with variable coefficients to be three order linear differential equation with constant coefficients.

The estimation of unknown function in a class of integro-sum inequalities has been proved by inductive approach and methodology of inequalities.