O Show that the Cauchy-Euler eigenvalue problem- equation (1) x^{2} \frac{d^{2} u}{d x^{2}}+x \frac{d u}{d x}+k^{2} u=0, \quad y(1)=y\left(e^{\pi}\right)=0 has eigenvalues given by k = n, n=1,2,... and eigenfunctions u(x) = sin (n ln(x)) (b) Put equation (1) into Sturm-Liouville form and thus identify the weight function[3 marks]w(x). Explain how we know that {sin ((n ln(x)), n = 1,2,...} is orthogonal over [1, e^] with-respect to the weight function found in part (b). (d) Find the corresponding generalised Fourier series for the function f(x): = x.Hint: in the integrals that you will have to solve, change the variable to 0 = ln(x)and use the maths handbook[11 marks]