I am working through Coddington ODEs. I have come across the following problem:
Determine all complex numbers $\ell$ for which the problem
$-y'' = \ell y$, $y(0) = 0$, $y(1) = 0$
has a non-trivial solutions, and compute such a solution for each these $\ell$.
My work so far is:
I found the characteristic equation to be
$r^2 + \ell = 0$
This implies that $r_1 = i\sqrt{\ell}$ and $r_2 = -i\sqrt{\ell}$.
This gives a general solution of $y = C_1 \cos{x\sqrt{\ell}} + C_2 \sin{x\sqrt{\ell}}$.
After plugging in the initial conditions, I found that $C_1 = 0$ and $C_2 = \frac{1}{\sin{\sqrt{\ell}}}$
From this, I concluded that this is valid for $\ell \ne k^2 \pi^2$ for integer $k$.
However, the answer in the book says that $\ell = n^2 \pi^2$ for natural numbers $n$ and that $y_n = \sin{n\pi x}$. I can't seem to see why this is true.