构造可积非自治二维线性微分方程组的一种新方法

文中建立的定理对求可积的非自治二维线性微分方程组提出了一种新方法 .在相当弱的条件下 ,用非奇异线性变换将方程组化为具斜对角系数矩阵的新方程组 ,从而把可积性判定归结到某个变系数二阶线性微分方程的讨论 .由选取后者为已知可积形式 ,并适当选取方程组的系数函数 ,即可导出许多新的可积非自治二维线性微分方程组 .

A new approach for constructing integrable two-dimensional systems of non-autonomuous linear differential equations

An approach for constructing new integrable two-dimensional systems of linear non-autonomous differential equations is proposed via the theorem obtained. Under some quite weak conditions,such systems are reduced to linear systems with a skew diagonal coefficient function matrix, and then the integrability investigation of the systems are transmited to some equivalent non-autonomous linear differential equations of the second order. New integrable two-dimensional linear non-autonomous differential systems are then obtained by a suitable choice of some integrable second order linear differential equation from the literature and the possible coefficient functions of the system under consideration such that the above-mentioned equivalent second order equation coincides with the chosen equation . Then the two-dimensional linear non-autonomous differential system composed of these coefficient functions should be integrable.