Question on damping string vibration.

I cannot derivate the formula of . Please read the question at the bottom and look at the way how I derivate the coeficients. You can see the formula don't match.

The original question is:

Find the solution of the equation for damping vibration string stretched from x=0 to x=L where

With boundary and initial condition given:

We assume

Attempted steps:

Using constant coeficients ODE,

Three cases of

For n=1,2,3.......

1)

(10)

2)

Question: For

The text book gave this equation:

I tried and I cannot arrive to this formula no matter what. Please look at (10) above.

It look very similar to the formula for .

But the book seldom make mistake!!! What did I do wrong?