I came across a problem that asked me to find the eigenpairs of the operator $M : C^2_D[0, \ell] \to C[0, \ell]$ defined by $$ My = -c\frac{d^2y}{dx^2}+dy,\ \ \ c,d > 0 $$ subject to Dirichlet boundary conditions $y(0) = 0, y(\ell) = 0$. So right away I attempted to solve $$ -c\frac{d^2y}{dx^2}+dy = \lambda y $$ This differential equation has the characteristic polynomial $$ -cr^2 + d - \lambda $$ with roots $r = \pm \sqrt{(d-\lambda)/c}$. So I plug them into my general solution: $$ y(x) = Ae^{\sqrt{(d-\lambda)/c}x} + Be^{-\sqrt{(d-\lambda)/c}x} $$ (Note: the $x$ is outside the sqrt)
Using the first boundary condition $y(0) = 0$, I see that $B = -A$. So I get the function $$ y(x) = Ae^{\sqrt{(d-\lambda)/c}x} - Ae^{-\sqrt{(d-\lambda)/c}x}. $$ Using the 2nd boundary condition I get $$ y(\ell) = Ae^{\sqrt{(d-\lambda)/c}\ell} - Ae^{-\sqrt{(d-\lambda)/c}\ell} = 0 $$ (Note: the $\ell$ is outside the sqrt)
This equation can hold only if $\lambda = d$, but if $\lambda = d$ then if I were to plug $\lambda$ back into $y(x)$ then I would get the zero function... $$ y(x) = Ae^{\sqrt{(d-d)/c}x} - Ae^{-\sqrt{(d-d)/c}x} $$ $$ y(x) = A - A = 0 $$ and the eigenfunction cannot be the zero function. Should I use a different method for finding the eigenvalues of $M$? Am I missing something basic in finding the eigenvalues?