$(30$ points$)$ The figure shows a thin beam carrying a uniformly distributed transverse loading on its top surface. The elasticity solution for this beam, in absence of body forces, gives the stress distribution as follows:

${}_{}=\frac{3w}{4c}(\frac{l^2}{c^2}-\frac{2}{5})y-A(x^{2}y-\frac{2}{3}{y}^3)$

${}_{yy}=-\frac{w}{2}+By+D{y}^3$

$($a$)$ Using ${}_{}$ and ${}_{yy}$ derive an expression for stress function $.$

$($b$)$ Determine ${}_{xy},$

$($c$)$ Determine the constants $A,B$ and $D($Hint: use equilibrium equations and boundary conditions for lower edge$).$