Find the eigenvalues and Eigen functions for homogenous two point boundary value problem
$$u''+\lambda u =0$$$$u'(0)=u'(1)=0$$$$\lambda> 0,u=A\cos \sqrt{\lambda}x+B\sin \sqrt{\lambda}x,$$$$B=0,\sqrt{\lambda}=n\pi,\lambda=n^2\pi^2$$$$cos{n{\pi}x}$$$$\lambda=0,u=A+Bx, B=0$$$$\lambda< 0,$$
not an eigenvalue:
$$u=Ae^{x\sqrt{-\lambda}}-Be^{-x{\sqrt{-\lambda}}},$$$$A=B=0$$
But where to from here?