Consider the following eigenvalue problem (a) Prove that the eigenvalue problem is self-adjoint and positive definite. (b) Show that the Eigen functions form an orthogonal set of functions in the...

Consider the following eigenvalue problem

(a) Prove that the eigenvalue problem is self-adjoint and positive definite.

(b) Show that the Eigen functions form an orthogonal set of functions in the interval [0, 1].

(c) Develop the function f r( )= 1 in a series of these Eigen functions.

(d) Substitute Un = hn=r into the differential equation for the Eigen functions.

Which differential equation do you obtain for the functions hn? What are the resulting boundary conditions?

(e) Obtain the general solution of the eigenvalues problem. Which equation do you obtain for the eigenvalues?

(f) Determine with the Eigen functions from (e) the integrals, which appeared in section (c), and give the series for f (r)= 1: