Question

Find the eigenvalues and eigenfunctions of

𝑦

′′ +



𝑦 = 0 , 0 < 𝑥 < 3

𝑦′(0) = 0 , 𝑦′(3) = 0

Answer 1

$$y''+\lambda y=0, y'(0)=0,y'(3)=0$$

The auxiliary equation

$$r^2 +\lambda=0$$

$$r=\pm\sqrt{-\lambda}, \lambda\leq0$$

The corresponding solution to our ODE will be

$$y=A\cos(\sqrt{\lambda }x)+B\sin(\sqrt{\lambda }x)$$

$$y'=-A\sqrt{\lambda}\sin(\sqrt{\lambda }x)+B\sqrt{\lambda}\cos(\sqrt{\lambda }x)$$

$$y'(0)=0=>B=0$$

$$y'(3)=0=>\sin(3\sqrt{\lambda })=0$$

$$3\sqrt{\lambda }=\pi n, n=1, 2, 3,...$$

$$\lambda_n=\dfrac{n^2\pi ^2}{9},n=1, 2, 3,...$$

The eigenvalue problem has the eigenvalue $\lambda_0=0,$ with associated eigenfunction $y_0=1,$ and infinitely many positive eigenvalues $\lambda_n=\dfrac{n^2\pi ^2}{9},$ with associated eigenfunctions $y_n=\cos(\dfrac{n\pi x}{3}), n=1,2,3,...$

There are no other eigenvalues.