In this problem we consider a higher order eigenvalue problem. In the study of
transverse vibrations of a uniform elastic bar one is led to the differential
equation
$$
y^{\mathrm{w}}-\lambda y=0
$$
$$
\begin{array}{l}{\text { where } y \text { is the transverse displacement and
} \lambda=m \omega^{2} / E I ; m \text { is the mass per unit length of }} \\\
{\text { the rod, } E \text { is Young's modulus, } I \text { is the moment of
inertia of the cross section about an }} \\ {\text { axis through the
centroid perpendicular to the plane of vibration, and } \omega \text { is the
frequency of }} \\ {\text { vibration. Thus for a bar whose material and
geometric properties are given, the eigenvalues }} \\ {\text { determine the
natural frequencies of vibration. Boundary conditions at each end are usually
}} \\ {\text { one of the following types: }}\end{array}
$$
$$
\begin{aligned} y=y^{\prime} &=0, \quad \text { clamped end } \\ y=y^{\prime
\prime} &=0, \quad \text { simply supported or hinged end, } \\ y^{\prime
\prime}=y^{\prime \prime \prime} &=0, \quad \text { free end } \end{aligned}
$$
$$
\begin{array}{l}{\text { For each of the following three cases find the form
of the eigenfunctions and the equation }} \\ {\text { satisfied by the
eigenvalues of this fourth order boundary value problem. Determine }
\lambda_{1} \text { and }} \\ {\lambda_{2}, \text { the two eigenvalues of
smallest magnitude. Assume that the eigenvalues are real and }} \\ {\text {
positive. }}\end{array}
$$
$$
\begin{array}{ll}{\text { (a) } y(0)=y^{\prime \prime}(0)=0,} &
{y(L)=y^{\prime \prime}(L)=0} \\ {\text { (b) } y(0)=y^{\prime \prime}(0)=0,}
& {y(L)=y^{\prime \prime}(L)=0} \\ {\text { (c) } y(0)=y^{\prime}(0)=0,} &
{y^{\prime \prime}(L)=y^{\prime \prime \prime}(L)=0 \quad \text {
(cantilevered bar) }}\end{array}
$$
