For the beam shown below:

$($a$)$ Develop expressions for shear, bending moment, slope and deflection by using double integration method.

$($b$)$ Write down the boundary conditions assuming the local coordinate systems at the left pin $(($:$x_1\}$ and at the $12kN$ load $(x_2).$

$($c$)$ Express the boundary conditions as a system of linear equations.

$($d$)$ Given that $EI=8,325kN-m^2,$ solve for the unknowns $($constants of integration$)$ using the matrix method.

$($e$)$ Based on the developed expression for deflection, determine the maximum deflection.