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Eigenvalue Problem : Transcendental Equation, Positive-Definite and Orthonormal

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Solve the eigenvalue problem

as follows:
Let U = ... be a two-component vector whose first component is a twice differentiable function u(x), and whose second component is a real number u1 Consider the corresponding vector space H with inner product

Let S C H be the subspace
....

and let

....
The above eigenvalue problem can now be rewritten in standard form LU=)U with U E S.
(a) PROVE or DISPROVE that L is self adjoint, i.e. that (V, LU> = (LV, U).
(b) PROVE or DISPROVE that L is positive-definite, i.e. that (U, LU) > 0 for U does not equal 0.
(c) FIND the (transcendental) equation for the eigenvalues of L.
(d) Denoting these eigenvalues by )1, )2, )3, . . . , EXHIBIT the orthonormalized eigenvectors U, n = 1, 2, 3,. .., associated with these eigenvalues.

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