Answer to Question #140443 in Differential Equations for Ruba

Question #140443

Find the eigenvalues and eigenfunctions of

𝑦

′′ +

𝑦 = 0 , 0 < 𝑥 < 3

𝑦′(0) = 0 , 𝑦′(3) = 0


1
Expert's answer
2020-10-27T19:18:52-0400
y+λy=0,y(0)=0,y(3)=0y''+\lambda y=0, y'(0)=0,y'(3)=0


The auxiliary equation


r2+λ=0r^2 +\lambda=0

r=±λ,λ0r=\pm\sqrt{-\lambda}, \lambda\leq0

The corresponding solution to our ODE will be


y=Acos(λx)+Bsin(λx)y=A\cos(\sqrt{\lambda }x)+B\sin(\sqrt{\lambda }x)

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

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

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

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

λn=n2π29,n=1,2,3,...\lambda_n=\dfrac{n^2\pi ^2}{9},n=1, 2, 3,...

The eigenvalue problem has the eigenvalue λ0=0,\lambda_0=0, with associated eigenfunction y0=1,y_0=1, and infinitely many positive eigenvalues λn=n2π29,\lambda_n=\dfrac{n^2\pi ^2}{9}, with associated eigenfunctions yn=cos(nπx3),n=1,2,3,...y_n=\cos(\dfrac{n\pi x}{3}), n=1,2,3,...

There are no other eigenvalues.



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