Two blocks of masses \(m\) and \(2m\) are placed on a frictionless horizontal surface and connected by an ideal spring of force constant \(k\) and relaxed length \(L_0\).

The blocks are pulled apart such that the total length of the spring becomes \(L_0 + \Delta x\), and the system is released from rest at time \(t = 0\). Let \(x(t)\) denote the displacement of the spring from its relaxed length at time \(t\). Which of the following differential equations correctly describes the motion of the system in terms of \(x(t)\), and what is the angular frequency \(\omega\) of the resulting oscillation?