Answer to Question #140443 in Differential Equations for Ruba

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.