$1-$ Show that the Airy function

$=\frac{3P}{4c}(xy-\frac{x{y}^3}{3c^2})+\frac{N}{4c}{y}^2$

solves the following cantilever beam problem, as shown in the following figure. As usual for such problems, boundary conditions at the ends $(x=0$ and $L)$ should be formulated only in terms of the resultant force system, while at $y=+-c$ the exact pointwise specification should be used. For the case with $N=0,$ compare the elasticity stress field with the corresponding results from strength of materials theory

Solve and show all steps step by step and show the derivation