关于二阶变系数线性方程的求解

ON THE FINDING OF SOLUTIONS FOR SECOND ORDER DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

给出了二阶变系数线性方程ｙ＂＋ｐ（ｘ）ｙ＇＋Ｑ（ｘ）ｙ＝０在可积的条件下的几种不同的求解方程，从而说明了这种可积的二阶方程不仅能用初等积分法求解而且也可以把它代数化，并使其通解公式化。同时也得到了二阶齐次变系数线性方程可积的其它一些充要条件。最后还讨论了当系数为周期函数时方程存在周期解的条件。

In §2, four met0hods for solving the integrable second order homogeneous linear ordinary differential equation with variable coefficientsy' +p(x)y'+Q(x)y=0 are given. Thus, the equation can be solved by means of elementary integrals and algebra. On the other hand, four necessary and sufficient conditions under which the equation is integrable are given. In §3, the sufficient conditions under which the homogeneous linear differential equation with periodic confficients possess nontrivial periodic solutions are given.