Abstract: The solvability of the second order linear differential equation with variable coefficients y″+P(x)y′+Q(x)y=f(x) is studied. A solvable necessary and sufficient condition: P(x)=bG(x)-G′(x)/G(x)-2F(x)，Q(x)=F²(x)-F′(x)-F(x)(bG(x)-G′(x)/G(x))+cG²(x). (F(x), G(x)∈C¹, G(x)≠0, b and c is constant) is obtained. Moreover, some solving processes and formula for teh general solution are given.

Key words: second order linear differential equation with variable coefficients, solvability, necessary and sufficient condition, solving process, general solution