A cart of initial mass \(m_0\) is attached to an ideal spring with spring constant \(k_0\) on a frictionless horizontal track. At time \(t = 0\), the cart is displaced from its equilibrium position \(x = 0\) and released from rest. As the cart oscillates, sand drops vertically into the cart from a stationary funnel above at a rate such that the total mass of the cart increases according to \(m(t) = m_0 e^{bt}\), where \(b\) is a positive constant. The falling sand has zero horizontal velocity immediately before entering the cart. Which of the following differential equations correctly governs the horizontal position \(x(t)\) of the cart as a function of time \(t\)?