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?