I am trying to follow along in a book by following an example that the author uses. However, I am completely stuck on one thing. I can't understand how the author got $A$ and $C$ to equal what they do (Eqs 4.6.18) from the system of equations (Eqs 4.6.9). I know how to find eigenvectors for regular systems of equations, but I haven't experienced this kind before. I've tried looking at it different ways and have put it into Mathematica, however, everything shows $A$ and $C$ to both be $\mathbf{0}$. Any help in figuring it out would be appreciated.
https://i.stack.imgur.com/13fL1.jpg
$A \mu sin(\mu h)+C[\mu h cos(\mu h)+\frac{2}{1+v} sin(\mu h)]=0$ $A \mu cos(\mu h)+C[-\mu h sin(\mu h)+\frac{1-v}{1+v} cos(\mu h)]=0$
$2 \mu h + sin(2 \mu h)=0$
$A = cos^2(\mu_{n} h) - \frac{2}{1+v}$
$C = \mu_{n}$
In the text, the author finds the eigenvalues first, and then plugs into the original equations to find the last two.