$3.$ Consider the simple structural problem described in the figure below.

The beam is homogenous $($shear modulus $\backslash (\backslash$mu $\backslash ))$ and uniform $($torsion constant J$).$ A distributed twisting moment $\backslash (\backslash$mathrm$\{$m$\}\_\{\backslash$mathrm$\{$t$\}\}(\backslash$mathrm$\{$x$\})\backslash )$ is applied along the whole span of the beam. It increases linearly with x starting from $0$ at $\backslash (\backslash$mathrm$\{$x$\}=0\backslash )$ to a value $\backslash (\backslash$mathrm$\{$m$\}\_\{\backslash$mathrm$\{$o$\}\}\backslash )$ at $\backslash (\backslash$mathrm$\{$x$\}=\backslash$mathrm$\{$L$\}\backslash ).$ It is applied in the direction indicated in the figure.

$($a$)$ Write down and solve the BVP describing the rotation $\backslash (\backslash$phi$($x$)\backslash )$ of the beam.

$($b$)$ Using Castigliano's second theorem, obtain the twist angle at the middle of the beam $(\backslash ($ x$=$L $/2)\backslash ).$ Indicate clearly in what direction the twisting will take place. Compare with the solution found in $($a$).$