Abstract: For linear ordinary differential equations, the analytical solution is convenient for qualitative analysis and practical application. However, most of the differential equations are not available in many cases. The regression method is applied to obtain approximate analytical solution, and the Least Square Support Vector Machine（LS-SVM） method is demonstrated the efficiency of the proposed method over existing methods. But this method requires not only to take the higher derivative of the kernel but also to solve a large linear equation system. Concerning higher order Ordinary Differential Equations（ODE）, the available approach is the reduction of the problem to a system of first-order differential equations and then the system is solved by LS-SVM regression method. These model parameters are adjusted to minimize an error function, finally, the high precision approximate solution（continuous and differentiable） is obtained by solving three system of linear equations. Experimental results verify the effectiveness of the proposed method.

Key words: approximate analytical solution, kernel function, least square support vector machine, ordinary differential equations, system of ordinary differential equations

摘要： 对于线性常微分方程，解析解方便定性分析和实际应用，然而大多数微分方程没有解析解。回归的方法被应用获取近似解析解，其中最小二乘支持向量机（LS-SVM）是目前为止最好的方法。但是该方法不仅需要对核函数求高阶导数而且需要求解一个大的线性方程组。为此，把高阶线性常微分方程转化为一阶线性常微分方程组，构建含有一阶导数形式的LS-SVM回归模型。该模型利用最小化误差函数去获得合适的参数，最终通过求解三个小的线性方程组获得高精度的近似解（连续、可微）。实验结果验证了该方法的有效性。

关键词: 近似解, 核函数, 最小二乘支持向量机, 常微分方程, 常微分方程组