In Section 4.2 it was shown that the displacement u of a nonuniform string satisfies...

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?