Problem 3.7 (a) Suppose that f(x) and g(x) are two eigenfunctions of an operator ộ, with the same eigenvalue q. Show that any linear combination of fand g is itself an eigenfunction of ô, with eigenvalue q. (b) Check that f (x) = exp (x) and g(x) = exp (-x) are eigenfunctions of the operator d² /dx?, with the same eigenvalue. Construct two linear combinations of fand g that are orthogonal eigenfunctions on the interval (-1, 1).

Problem 3.7 (a) Suppose that f(x) and g(x) are two eigenfunctions of an operator ộ, with the same eigenvalue q. Show that any linear combination of fand g is itself an eigenfunction of ô, with eigenvalue q. (b) Check that f (x) = exp (x) and g(x) = exp (-x) are eigenfunctions of the operator d² /dx?, with the same eigenvalue. Construct two linear combinations of fand g that are orthogonal eigenfunctions on the interval (-1, 1).