The piston$-$pendulum illustrated is a two degree$-$of$-$freedom system where $q_1$ and $q_2$ are defined to be $x$ and $$ respectively. The cart has a mass of $m_1$ and the pendulum has a mass of $m_2$ with a length of l $.$

$($a$)$ Using the Lagrangian, derive the coupled equations of motion for the system in symbolic terms. Write the equations of motion in matrix form.

$L=T-U,\frac{d}{dt}[\frac{\text{d}\text{e}\text{l}\text{L}}{del{q}_i^{}}]-\frac{\text{d}\text{e}\text{l}\text{L}}{del{q}_i}=0$

Please solve the Lagrangian in terms of x and $$