Eigenvalues and eigenfunctions

Find the eigenvalues and eigenfunctions for

$\displaystyle y''+\lambda y=0$, $\displaystyle 0\leq x\leq1$

with the homogenous boundary conditions $\displaystyle y(0)=y'(1)=0$.

So I subbed in $\displaystyle y=e^{\alpha x}$ et cetera, and somewhere down the line ended up needing to solve

$\displaystyle \sin{\omega}+\omega\cos{\omega}=0$

and now I'm stuck and not even sure if this can be done. Obviously $\displaystyle \omega=0$ would work, but wouldn't that just give a trivial solution?