(a) Suppose that f(x) and g(x) are two eigenfunctions of an operator , with the same eigen...

(a) Suppose that f(x) and g(x) are two eigenfunctions of an operator , with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfunction of , with eigenvalue q.

(b) Check that f(x) = exp(x) and g(x) = exp(–x) are eigenfunctions of the operator d2/dx2, with the same eigenvalue. Construct two linear combinations of f and g that are orthogonal eigenfunctions on the interval (–1, 1).