Consider the boundary value problem

$-y^{\text{'}\text{'}}=x$

$y(0)=y(1)=0$

$(10$ pts$)$ Find the exact solution of the problem.

$(10$ pts$)$ Derive the weak form of the problem.

$(20pts)$ For the Galerkin approximation of the problem, let $N=3$ and choose the basis function ${}_i=sin(ix),i=1,2,3.$ Calculate the stiffness matrix, $K_{ij}$ and the load vector $f_i.$ Solve for the coefficients and construct the approximate solution $y_h.$ Plot the exact and approximate solutions.

$(40$ pts$)$ Using a uniform mesh with mesh size $h=0\mathrm{.}25,$ compute the Galerkin piece$-$wise linear finite element approximation by hand. Plot the exact and approximate solutions.