# Decomposition of 2 fourth order differential equations on 4 equations of second order

How to make decomposition of 2 fourth order differential equations on 4 equations of second order

$$C_1x''''+C_2x'''+C_3x''+C_4x'+C_5x-C_6y''-C_7y=0$$

$$D_1y''''+D_2y'''+D_3y''+D_4y'+D_5y-D_6x''-D_7x=0$$

Known constants $$C_i, D_i$$

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Define $w=x''$ and $v=y''$.
Write your system in the matrix form $\bf{z}^{\prime}=A\mathbf{z},$ where $\bf{z}=\left( x,x^{\prime},x^{\prime\prime},x^{\prime\prime\prime },y,y^{\prime},y^{\prime\prime},y^{\prime\prime\prime}\right) ^{T}$ (you can divide the first equation with $C_{1}$ and the second one with $D_{1}%$, assuming these are non-zero). Now try to find a transformation $\bf{z=}T\mathbf{q}$ such that $\bf{q}^{\prime}=T^{-1}AT\mathbf{q}$ has the required form (here $\mathbf{q}=\left( q_{1},q_{1}^{\prime},q_{2},q_{2}^{\prime},q_{3},q_{3}^{\prime},q_{4},q_{4}^{\prime}\right) ^{T}$ with your notation). Is this possible?