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Trying to do this one...

A model for the transverse vibrations of a stretched string with variable density ρ and tension τ (both continuous and strictly positive on the closed interval [0,l]):

PDE: $ρ(x)u_{tt} − [τ (x)u_x]_x = 0, 0 < x < l, t > 0,$

BC: $u(0, t) = 0, u_x(l, t) + A_l u(l, t) = 0 \quad(for \quad all \quad t > 0)$

One end of the string is fixed, and at the other end the string exchanges some energy with the endpoint). If u(x, t) = X(x)T(t) is to be a separated solution of the PDE, find the ODEs that T(t) and X(x) must satisfy. If the BC are also to be satisfied, verify that X(x) must then be a solution of a Sturm-Liouville problem. In particular, verify explicitly that the boundary conditions for the boundary-value problem for X(x) are symmetric.

So

if $ u(x, t) = X(x)T(t) $

$u_{tt} = X(x)T''(t) $

$u_{x} = X'(x)T(t)$

$u_{xx} = X''(x)T(t)$

So:

PDE: $ρ(x)u_{tt} − [τ (x)u_x]_x = ρ(x)u_{tt} - τ_x(x)u_x- τ(x)u_{xx} $

$= ρ(x)X(x)T''(t) - τ_x(x)X'(x)T(t) - τ(x)X''(x)T(t)$

$\rightarrow ρ(x)X(x)T''(t) = (τ_x(x)X'(x) + τ(x)X''(x))T(t)$

$\rightarrow \frac{T''(t)}{T(t)} = \frac{τ_x(x)X'(x) + τ(x)X''(x)}{ρ(x)X(x)} = -\lambda$

$(1)T''(t)+\lambda T(t) = 0 $

$(2) τ(x)X''(x) + τ_x(x)X'(x) +\lambda ρ(x)X(x)=0$

I think up to this point is correct? Not 100% confident though.

So I need to answer these questions:

(a) Find the ODE's that T(t) and X(x) must satisfy - I think the ODE's are (1) (2) above?

(b) If the BC are also satisfied, verify X(x) must be a solution of a S-L problem.

(c)In particular, verify that the boundary conditions for the boundary - value problem for X(x) are symmetric.

Think I got (a) but stuck on (b) and (c)

UPDATE:

I got (b)

$(τ(x)X'(x))' + \lambda ρ(x)X(x) = 0$

but still confused about (c) and where exactly to use the BC's

Any help would be much appreciated. Just need help with this last little bit - verify that the boundary conditions for the boundary - value problem for X(x) are symmetric.

thanks!

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