线性常微分方程的微分转移矩阵法解析解
Analytical solution of linear ordinary differential equations by differential transfer matrix method
摘要：
本文给出了任意阶变系数齐次线性常微分方程精确解的一种新的解析方法。该方法是基于跳跃转移矩阵的定义和扩展到极限微分形式。该方法将$n$阶微分方程化为单位阶的$n$线性微分方程组。完整的解析解，然后发现由微扰技术。所提出的方法的重要特点是，它涉及的独立的解决方案，而不是其衍生物的演变。通过直接代入原微分方程的解，证明了方法的有效性。我们讨论了微分转移矩阵的一般性质，并给出了几个分析例子，显示了该方法的适用性。我们表明，Abel-Liouville-Ostogradski定理可以很容易地恢复通过这种方法。
英文摘要：
We report a new analytical method for exact solution of homogeneous linear ordinary differential equations with arbitrary order and variable coefficients. The method is based on the definition of jump transfer matrices and their extension into limiting differential form. The approach reduces the $n$th-order differential equation to a system of $n$ linear differential equations with unity order. The full analytical solution is then found by the perturbation technique. The important feature of the presented method is that it deals with the evolution of independent solutions, rather than its derivatives. We prove the validity of method by direct substitution of the solution in the original differential equation. We discuss the general properties of differential transfer matrices and present several analytical examples, showing the applicability of the method. We show that the Abel-Liouville-Ostogradski theorem can be easily recovered through this approach.
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