It is well-known that one can start with a second-order ODE and re-write it as a system of two first-order ODEs.
I have the below pair of coupled non-linear second-order ODEs for $x$ and $y$
$\gamma_1 \dot{x} - \gamma_2 ( \dot{x} - \dot{y} ) = (m_1 + m_2) d^2 \ddot{x} + I \ddot{x} + m_2 d^2 \ddot{y} + d(m_1 + m_2) g x,$
$\gamma_2 (\dot{x}- \dot{y}) = m_2 d^2 \ddot{y} + I \ddot{y} + m_2 d^2 \ddot{x} + dm_2g y, $
where $\gamma_1$, $\gamma_2$, $m_1$, $m_2$, $g$, $d$ and $I$ are parameters of the system.
Is it possible to convert this into a system of four coupled non-linear first-order ODEs?