# Decoupling a system of two second-order differential equations

Suppose I had a system of differential equations \begin{align} x''(t) + A_1(t) x'(t) + B_1(t) x(t) + C_1(t) y(t) &=0\\ y''(t) + A_2(t) y'(t) + B_2(t) y(t) + C_2(t) x(t) &=0, \end{align} where we want to solve for $$x(t)$$ and $$y(t)$$. Here, $$A_1, A_2, B_1, B_2, C_1, C_2$$ are some coefficient functions of the independent variable $$t$$.

Is there a way to decouple this by writing it as some type of matrix differential equation and diagonalizing? If this cannot be done for general functions $$A, B,C$$, then what if we take the case where all of those coefficient functions are constants? Can progress be made then?