Thanks for your help TSny, but I'm still a little lost.
I end up with three equations of the form
\dot{m}_x = f_1 (m_y,m_z,\dot{m}_y,\dot{m}_z) \\
\dot{m}_y = f_2 (m_x,m_z,\dot{m}_x,\dot{m}_z) \\
\dot{m}_z = f_3 (m_x,m_y,\dot{m}_x,\dot{m}_y) \\
but I don't see how to...