Chapter 17.1, Problem 33E

Let L be a nonzero real number.

(a) Show that the boundary-value problem $y^{″}+\text{λ}y=0$, $y\left(0\right)=0$, $y\left(L\right)=0$ has only the trivial solution $y=0$, for the cases $\text{λ}=0$ and $\text{λ}<0$.

(b) For the case $\text{λ}>0$, find the values of $\text{λ}$ for which this problem has a nontrivial solution and give the corresponding solution.