
The moment at any point point on the beam which is at distance x from the left support is
$M_x = M - Rx$
By double integration method
$EI \, y'' = M_x$
$EI \, y'' = M - Rx$
$EI \, y' = Mx - \frac{1}{2}Rx^2 + C_1$
$EI \, y = \frac{1}{2}Mx^2 - \frac{1}{6}Rx^3 + C_1x + C_2$
Boundary conditions
At x = 0, y = 0; C2 = 0
At x = L, y = 0;
$0 = \frac{1}{2}ML^2 - \frac{1}{6}RL^3 + C_1L$
$C_1 = \frac{1}{6}RL^2 - \frac{1}{2}ML$
At x = L, y' = 0;
$0 = ML - \frac{1}{2}RL^2 + (\frac{1}{6}RL^2 - \frac{1}{2}ML)$
$0 = M - \frac{1}{2}RL + \frac{1}{6}RL - \frac{1}{2}M$
$\frac{1}{3}RL = \frac{1}{2}M$
$R = \dfrac{3M}{2L}$ answer