$2.$ Two cantilever beams, sharing the same cross$-$sectional geometry and Young's modulus $(\backslash ($ E $\backslash )),$ are built$-$in as illustrated in Fig. P$2($a$)$ and subjected to an end load $\backslash ($ P $\backslash ),$ as shown in Fig. P$2($b$).$ It is assumed that the two beams remain in contact at $\backslash ($ x$=0\backslash )$ and $\backslash ($ x$=$L $\backslash )$ during the deflection of the beam.

$[25$ pts$]$

$($a$)$ Derive the equation of the elastic curve using the integration method. $[15$ pts$]$

$($b$)$ Determine the maximum deflection and the maximum slope of the beam. $[10$ pts$]$

Fig. P$2$