We concisely summarize a method of finding all rational solutions to an inhomogeneous rational ODE system of arbitrary order (but solvable for its highest order terms) by converting it into a finite dimensional linear algebra problem. This method is then used to solve the problem of conclusively deciding when certain rational ODE systems in upper triangular form can or cannot be reduced to diagonal form by differential operators with rational coefficients. As specific examples, we consider systems of coupled Regge-Wheeler equations, which have naturally appeared in previous work on vector and tensor perturbations on the Schwarzschild black hole spacetime. Our systematic approach reproduces and complements identities that have been previously found by trial and error methods.

我们简明地总结了一种方法，该方法通过将其转化为有限维线性代数问题来找到任意阶（但可求解其最高阶项）的非齐次有理ODE系统的所有有理解。然后，该方法用于解决最终确定某些上三角形式的有理ODE系统是否可以由具有有理系数的微分算子还原为对角线形式的问题。作为特定示例，我们考虑耦合Regge-Wheeler方程组的系统，这些系统自然出现在先前关于Schwarzschild黑洞时空的矢量和张量摄动的工作中。我们的系统方法再现并补充了以前通过反复试验方法发现的身份。