Problem $4(25$ points$).$ Consider the simply supported beam $AB$ shown below. The beam is subjected

to a concentrated force $P$ at point $C.EI$ is constant.

$($a$)$ Determine the expression for the elastic curve $($i$.$e$.,$ deflected shape $y(x))$ of the beam using the method

of double integration. Take the support $A$ as the origin of the $x-$axis.

$($b$)$ Using the results in part $($a$),$ find the maximum deflection in the beam and where it occurs.

$($c$)$ Using the results in part $($a$),$ find the Elopes at the supports A and $B.$