Problem $5-55$: Three masses are connected by massless pulleys and inextensible cables, as sketched in Figure P$5-55.$ Mass my is excited by an external force F $($t$).$ All the masses move without friction with respect to their constraining surfaces. Assume small motions about the equilibrium configuration of the system and assume F$($t$)$ is much smaller than the weight of m$,$ such that all the cables remain taut at all times, and note that gravity acts. Derive the equation$$s$$ of motion for the system. Begin by selecting a complete and independent set of generalised coordinates, starting with x$1$t$$ then x$2($t$$ and x$3($t$)$as necessary. Note that the meanings of these coordinates remain to be clearly defined.

ANS: attached too.

$(m_1+\frac{1}{4}{m}_3)x_1+\frac{1}{4}{m}_3{x}_2+\frac{1}{4}k{x}_1+\frac{1}{4}k{x}_2=F(t)-(m_1-\frac{1}{2}{m}_3)g$

$\frac{1}{4}{m}_3{x}_1+(m_2+\frac{1}{4}{m}_3)x_2+c{x}_2^{}+\frac{1}{4}k{x}_1+\frac{1}{4}k{x}_2=\frac{1}{2}{m}_{3}g$

Can you please show me in detail explanation on how to get the answeres. Process and thought processing too! Thanks.