Problem $1$: $P(x_l)w_i(x_l)+P(x_0)w_i(x_0)-{\displaystyle \underset{x_0}{\overset{x_l}{}}}EA\frac{du}{dx}\frac{d{w}_i}{dx}+{\displaystyle \underset{x_0}{\overset{x_l}{}}}q(x)w_i(x)dx=0$

A uniform bar is fixed at both ends and subjected to a distributed axial load $q(x)($ see drawing below$).$ The values defining the bar and load are: $A=500m{m}^2,E=100$MPa,$L=0\mathrm{.}6m,c=200k\frac{N}{m^4}($watch your units!$).$ Using the Galerkin method with an assumed quadratic polynomial solution, determine the axial force distribution $P(x)$ along the bar. Plot $P(x)$ for $0xL.$ Find the reaction forces at the left and right sides and the total force due to $q(x),$ and sum them to determine whether static equilibrium is satisfied. The given equation is the starting equation.