The cantilever beam shown below has constant $EI$ throughout its length.

a$)$ Show that the distributed loading $w(x)$ can be expressed using singularity functions as

$w(x)=w_0($:$x-2a$:${)}^0-\frac{w_0}{3a}($:$x-2a$:${)}^1$

b$)$ Determine the elastic curve $y(x)$ in terms of singularity functions.

c$)$ Determine the deflection at the free end of the beam.