Consider the boundary-value problem introduced in the construction of the mathematical model for the shape of a rotating string: T d2y dx2 + rhoω2y = 0, y(0) = 0, y(L) = 0. For constants T and rho, define the critical speeds of angular rotation ωn as the values of ω for which the boundary-value problem has nontrivial solutions. Find the critical speeds ωn and the corresponding deflections yn(x). (Give your answers in terms of n, making sure that each value of n corresponds to a unique critical speed.)

Explanation:

The given differential equation is

and y(0) = 0, y(L) =0

where T and ρ  are constants

The given rewrite as

auxiliary equation is

Solution of this de is

y(0)=0 ⇒ C₁ = 0

y(L) = 0 ⇒

we need non zero solution

⇒ C₂ ≠ 0 and

solution corresponding these values

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explanation:

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step-by-step explanation: