Consider the boundary-value problem introduced in the

Chapter 5, Problem 25E

(choose chapter or problem)

Consider the boundary-value problem introduced in the construction of the mathematical model for the shape of a rotating string:

\(T \frac{d^{2} y}{d x^{2}}+\rho \omega^{2} y=0, \quad y(0)=0, \quad y(L)=0\)

For constant T and \(\rho\), define the critical speeds of angular rotation \(\omega_{n}\) as the values of \(\omega\) v for which the boundary value problem has nontrivial solutions. Find the critical speeds \(\omega_{n}\) and the corresponding deflections \(y_{n}(x)\).

Text Transcription:

Tfracd^2ydx^2+rho\omega^2y=0,y(0)=0,y(L)=0

rho

omega_n

omega

y_n(x)