$L=12$;

$P=7$;

$q(x)=3Px(x-\frac{L}{4})$

State mathematical formulation of the problem $($balance equation and boundary conditions$)(3$ points$)$

Determine the type of boundary conditions $($essential or natural$)(2$ points$)$

Solve the problem by Galerkin method using ${}_1=N_1=x,{}_2=N_2=x^2$ points$).$

Compute the residual $r_f$ of the boundary condition specified at $x=L(2$ points$).$

Determine the residual in balance equation, $r_b*(2$ points$)$

Solution MUST CONTAIN all necessary formulas, equations and all intermediate results.

Direction of the distributed force q$($x$)$ and its sign are embedded into the equation of q$($x$)($this means that

you just put the expression for q$($x$)$ as is into the Galerkin equations$)$ you do not need to change its sign.

The concentrated force FL points upwards $($in the same direction as y$-$axis$)$

Please make sure all steps are clear and legible. Please don't use a previous chegg answer, work the problemme based on the picture, description and what it is asking for. I will upvote!!!!