Consider the eigenvalue problem

y'' + y' + y = 0,

,y(0) = y(1) = 0

(i) Write the problem in Sturm-Liouville form.

.(ii) Find the eigenvalues and eigenfunctions.

(iii)Find the orthogonal expansion of f(x) = x on [0, 1], in terms of the eigenfunctions you found in (ii).

(iv) ] Find the smallest N such that the weighted L²error between the Nth partial sum, FN (x) in (iii), and f(x) is less than 101..

Plot f and FN (x) on the same graph for this N value.