(Solved): The eigenvalues in this problem are all nonnegative. First determine whether =0 is an eigenvalue; ...

The eigenvalues in this problem are all nonnegative. First determine whether $?=0$ is an eigenvalue; then find the positive eigenvalues and associated eigenfunctions. $y^{??}+9?y=0;y^{?}(0)=0,y(1)=0$ Is $?=0$ an eigenvalue of this problem? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. No, when $?=0$ the only solution to the given equation is the trivial solution. B. Yes, $?=0$ is an eigenvalue with corresponding eigenfunction $y_0(x)=$ C. No, when $?=0$ there are an infinite number of nontrivial solutions to the given equation. What are the positive eigenvalues ${?}_n$ and associated eigenfunctions $y_n(x)$ for $n=1,2,3,\dots$ Select the correct choice below and fill in the answer boxes to complete your choice. (Type exact answers.) A. The positive eigenvalues are ${?}_n=$. The eigenfunctions are $y_n(x)=$ regardless of whether $n$ is even or odd. B. The positive eigenvalues are ${?}_n=$. The eigenfunctions are $y_n(x)=$ when $n$ is even and $y_n(x)=$ $n$ is odd.