Adjoint Operator (20 points) Derive the following properties of the adjoint of an operator:

SOLVED: For the very special case λ=0 and q(x)=0, the self-adjoint eigenvalue equation becomes (d)/(d x)[p(x) (d u(x))/(d x)]=0 satisfied by (d u)/(d x)=(1)/(p(x)) Use this to obtain a "second" solution of the following: (a) Legendre's equation; (b) Laguerre's equation; and (c) Hermite's equation. ANS. (a) u2(x)=(1)/(2)ln(1 x)/(1-x) (b) u2(x)-u2(x0)=∫x0^x e^t(d t)/(t) ; (c) u2(x)=∫0^x e^t^2 d t These second solutions illustrate the divergent behavior usually found in a second sol