[UPSC CSE 2001] Rigid Body Dynamics (Q233, 20M)

A system with 2 independent coordinates x_1 and x_2 has the following Larangian L = \frac{1}{2} \frac{(\dot{x}_1)^2}{\alpha + \beta x_2} + \frac{1}{2} (\dot{x}_2)^2 (x_2)^2 - \gamma x_2 \alpha, \beta, \gamma being constants. Obtain the Lagrangian equation of motion.