In Section 4.2 it was shown that the displacement u of a nonuniform string satisfies
where Q represents the vertical component of the body force per unit length. If Q = 0, the partial differential equation is homogeneous. A slightly different homogeneous equation occurs if Q = αu.
(a) Show that if α < 0, the body force is restoring (toward u = 0). Show that if α > 0, the body force tends to push the string further away from its unperturbed position u = 0.
(b) Separate variables if ρ0(x) and α(x) but T0 is constant for physical reasons.
Analyze the time-dependent ordinary differential equation.
*(c) Specialize part (b) to the constant coefficient case. Solve the initial value problem if α < 0:
What are the frequencies of vibration?
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