Question

In Problems 9 and 10 the eigenvalues and eigenfunctions of theboundary-value problem$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathit{\lambda }\mathit{y}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathit{\pi }\mathbf{\right)}\mathbf{=}\mathbf{0}$are${\mathit{\lambda }}_{\mathbf{n}}\mathbf{=}{\mathit{n}}^{\mathbf{2}}\mathbf{,}\mathit{n}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{.}\mathbf{..}\mathbf{,}$and$\mathit{y}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{n}\mathit{x}$, respectively. Fill in theblanks.

A solution of the BVP when$\mathit{\lambda }\mathbf{=}\mathbf{36}$is$\mathit{y}\mathbf{=}\mathbf{\_}\mathbf{\_}\mathbf{\_}\mathbf{\_}$because _____.

Short answer

A solution of the BVP when$\lambda =36$ is $y=\cos 6x$because $\lambda ={\left(6\right)}^2$is not an eigen-value.

Step-by-step solution

Step 1:Define Eigenvalues and Eigenvectors.

Simple harmonic motion is a type of periodic motion in mechanics and physics in which the restoring force on a moving object is directly proportional to the size of the object's displacement and acts in the direction of the object's equilibrium position.

Determine the solution of the BVP.

The eigen-values and eigen-functions of the given boundary value problem (BVP) are:

$\begin{array}{c}{\lambda }_n=0,1,4,9,16,25,36,49,64,\dots ,(\text{as}{\lambda }_n=n^2,n=0,1,2,3,4,\dots ,)\\ y=\cos nx\end{array}$

Hence, a solution of the BVP when$\lambda =36$is$y=\cos 6x$because$\lambda ={\left(6\right)}^2$is not an eigen-value.