An integrable theorem of second order linear non-homogeneous differential equation

The solvable of second order linear non-homogeneous differential equation which in important in (theory) and practice is studied.The concept of characteristic equation system is leaded into second order linear differential equation.Thus a new practical integrable sufficient condition that variable coefficient second order linear non-homogeneous differential equation and analytic expression fomula of the general solution is given.This extend the solution of classical constant coefficient second order linear non-homgeneous differential (equation).