I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman, but I have been impossible these paragraphs.
The displacement $u$ of a nonuniform string satisfies $$\rho_0(x)\frac{\partial^2 u}{\partial t^2} = T_0\,\frac{\partial^2 u}{\partial x^2} + Q(x,t),$$ 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 $Q = \alpha u$.
(a) Show that if $\alpha < 0$, the body force is restoring (toward $u= 0$). Show that if $\alpha > 0$, the body force tends to push the string further away from its unperturbed position $u = 0$.
(b) Separate variables if $\rho_o(x)$ and $\alpha(x)$ but $T_o$ 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 $\alpha < 0$:
\begin{align*} u(0,t)=0&&u(x,0)=0\\ u(L,t)=0&&\frac{\partial u}{\partial t}(x,0)=f(x). \end{align*}
What are the frequencies of vibration?
I have this but I need the analysis of the temporal part and the whole first paragraph
(b)-(c)By sepration of variables, $u=\phi(x)h(t)$, $\frac{d^2h}{dt^2}=-\lambda h$ and $T_0\frac{d^2\phi}{dx^2}+(\alpha+\lambda\rho_0)\phi=0$ whit the bundary conditions $\sqrt{(\alpha+\lambda\rho_0)/T_0}=n\pi/L$, and $\phi=\sin(n\pi x/L)$. In general $h(t)$ involves a linear combination of $\sin\sqrt\lambda t$ and $\cos\sqrt\lambda t$, but the homogeneous initial condition $u(x,0)=0$ implies there are no cosines. THus by superposition $$u(x,t)=\sum_n A_n \sin \sqrt{\lambda_n}t\sin(n\pi x/L),$$ where the frecuincies of vibration are $\sqrt{\lambda_n}=\sqrt{\frac{(n\pi/L)^2T_0-\alpha}{\rho_0}}.$