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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?

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