$(1)$
対称性より\[
R_A=R_B=\frac{1}{2}ql
\]
$(2)$
\[\begin{align}
M
&=R_Ax-\frac{1}{2}qx^2-M_A\\
&=\frac{1}{2}qlx-\frac{1}{2}qx^2-M_A
\end{align}\]
$(3)$
\[
EI\theta=\frac{1}{6}qx^3-\frac{1}{4}qlx^2+M_Ax+C_1\\
EI\omega =\frac{1}{24}qx^4-\frac{1}{12}qlx^3+\frac{1}{2}M_Ax^2+C_1x+C_2\]
\[
x=0で\theta=0よりC_1=0\\
x=0で\omega =0よりC_2=0
\]
\[
\theta=\frac{1}{EI}\left(\frac{1}{6}qx^3-\frac{1}{4}qlx^2+M_Ax\right)\\
x=\frac{1}{2}l \ で \ \theta=0より
\]
\[\begin{align}
\theta\big|_{x=\frac{1}{2}l}
&=\frac{1}{EI}\left\{\frac{1}{6}q\left(\frac{l}{2}\right)^3-\frac{1}{4}ql\left(\frac{l}\\
{2}\right)^2+M_A\left(\frac{l}{2}\right)\right\}\\
&=0
\end{align}\]
\[
\therefore M_A=\frac{1}{12}ql^2
\]
$(4)$
\[
\omega =\frac{1}{EI}\left(\frac{1}{24}qx^4-\frac{1}{12}qlx^3+\frac{1}{24}ql^2x^2\right)
\]
$(5)$
\[
x=\frac{1}{2}lを代入する
\]
\[\begin{align}
\omega\big|_{x=\frac{1}{2}l}
&=\frac{1}{EI}\left\{\frac{1}{24}q\left(\frac{l}{2}\right)^4-\frac{1}{12}ql\left(\frac{l}
{2}\right)^3+\frac{1}{24}ql^2\left(\frac{l}{2}\right)^2\right\}\\
&=\frac{ql^4}{384EI}
\end{align}\]