y ′ ′ + λ y = 0 , y ′ ( 0 ) = 0 , y ′ ( 3 ) = 0 y''+\lambda y=0, y'(0)=0,y'(3)=0 y ′′ + λ y = 0 , y ′ ( 0 ) = 0 , y ′ ( 3 ) = 0
The auxiliary equation
r 2 + λ = 0 r^2 +\lambda=0 r 2 + λ = 0
r = ± − λ , λ ≤ 0 r=\pm\sqrt{-\lambda}, \lambda\leq0 r = ± − λ , λ ≤ 0 The corresponding solution to our ODE will be
y = A cos ( λ x ) + B sin ( λ x ) y=A\cos(\sqrt{\lambda }x)+B\sin(\sqrt{\lambda }x) y = A cos ( λ x ) + B sin ( λ x )
y ′ = − A λ sin ( λ x ) + B λ cos ( λ x ) y'=-A\sqrt{\lambda}\sin(\sqrt{\lambda }x)+B\sqrt{\lambda}\cos(\sqrt{\lambda }x) y ′ = − A λ sin ( λ x ) + B λ cos ( λ x )
y ′ ( 0 ) = 0 = > B = 0 y'(0)=0=>B=0 y ′ ( 0 ) = 0 => B = 0
y ′ ( 3 ) = 0 = > sin ( 3 λ ) = 0 y'(3)=0=>\sin(3\sqrt{\lambda })=0 y ′ ( 3 ) = 0 => sin ( 3 λ ) = 0
3 λ = π n , n = 1 , 2 , 3 , . . . 3\sqrt{\lambda }=\pi n, n=1, 2, 3,... 3 λ = πn , n = 1 , 2 , 3 , ...
λ n = n 2 π 2 9 , n = 1 , 2 , 3 , . . . \lambda_n=\dfrac{n^2\pi ^2}{9},n=1, 2, 3,... λ n = 9 n 2 π 2 , n = 1 , 2 , 3 , ...
The eigenvalue problem has the eigenvalue λ 0 = 0 , \lambda_0=0, λ 0 = 0 , with associated eigenfunction y 0 = 1 , y_0=1, y 0 = 1 , and infinitely many positive eigenvalues λ n = n 2 π 2 9 , \lambda_n=\dfrac{n^2\pi ^2}{9}, λ n = 9 n 2 π 2 , with associated eigenfunctions y n = cos ( n π x 3 ) , n = 1 , 2 , 3 , . . . y_n=\cos(\dfrac{n\pi x}{3}), n=1,2,3,... y n = cos ( 3 nπ x ) , n = 1 , 2 , 3 , ...
There are no other eigenvalues.
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