Consider the boundary-value problem introduced in the construction of the mathematical model for the shape of a rotating string: For constant T and ρ, define the critical speeds of angular problem rotation ωn as the values of ω for which the boundary-value has nontrivial solutions. Find the critical speeds ωn and the corresponding deflections yR(x).

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We have (d^(2)y)/(dx^(2))+(rhow^(2))/(T)y=0Thus, y=c_(1)cos sqrt((rhow^(2))/(T))x+c_(2)sqrt((rhow^(2))/(T))xFrom y(0)=0 we get c_(1)=0And y(1)=0 implies 0=0*cos sqrt((rhow^(2))/(T))L+c_(2)sin sqrt((rhow^(2))/(T))L=>c_(2)sin sqrt((rhow^(2))/(T))L=0For nontrivial solution c_(2)!=0, so ... See the full answer