Chapter 17.1, Problem 33E

(a)

To determine

To show: The boundary-value problem $y^{″}+\lambda y=0$, $y\left(0\right)=0$, $y\left(L\right)=0$ has only the trivial solution $y=0$ for the case $\lambda =0$ and the boundary-value problem $y^{″}+\lambda y=0$, $y\left(0\right)=0$, $y\left(L\right)=0$ has only the trivial solution $y=0$ for the case $\lambda <0$.

(b)

To determine

To find: The values of $\lambda$ for boundary-value problem, which has a nontrivial solution and give the corresponding solution.