Consider the vibrations of a nonuniform string of mass density ρ0(x). Suppose that the left end at x = 0 is fixed and the right end obeys the elastic boundary condition: ∂u/∂x = −(k/T0)u at x = L. Suppose that the string is initially at rest with a known initial position f(x). Solve this initial value problem. (Hints: Assume that the appropriate eigenvalues and corresponding eigenfunctions are known. What differential equations with what boundary conditions do they satisfy? The eigenfunctions are orthogonal with what weighting function?)
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