Find the eigenvalue and eigenvector

Question

Can someone give me some guidance on how to obtain the eigenvalue and eigenvector for the following equation? Don't I need another equation?

$EIu_{yy}+pu=0$ where $\lambda=\frac{p}{EI}$

Answer 1

Write it as

$$ \frac{{\rm d}^2u}{{\rm d}y^2} = - \lambda u $$

And try solutions of the form

$$ u(y) = A e^{\pm i\sqrt{\lambda}y} $$

Eigenvectors are $u$ and eigenvalues are $\lambda$