Math Help - Separation of variables

1. Separation of variables

Hi, I have this problem, but I am stuck in one thing.

Assuming that solutions to $y_{tt}=g(zy_z)_z$ are of the form $y(z,t)=Z(z)T(t)$, and that the separation constant is negative, written as $-v^2$ with $v>0$, I need to show that $Z(z)$ and $T(t)$ must satisfy the differential equations

$T''+v^2gT=0$, and
$zZ''+Z'+v^2Z=0$

So well, I kind know what to do once I have substituted $y(z,t)$ into $y_{tt}=g(zy_z)_z$... However, I keep on getting the wrong results:

I get this:

$Z(z)T''(t)=g(Z(z)Z'(z)T(t))_z$
$Z(z)T''(t)=g_z(T(t)(Z'(z)Z'(z)+Z(z)Z''(z)))$

But this is clearly not what should be on both sides :/

I would really appreciate it if any of you could give me a hand. Once I get both results on each side, I will be able to separate the variables.

Thanks a lot!

2. Re: Separation of variables

$zy_z \neq Z(z)Z'(z)T(t)$ unless $Z(z) = z$.

I think you should have:

\begin{align*}Z(z)T''(t) & = g\left(zZ'(z)T(t)\right)_z \\ & = g'\left(zZ'(z)T(t)\right) \left[zZ''(t)T(t) + Z'(t)T(t)\right]\end{align*}

Also, I am assuming that $g$ is a function. If it is a constant, that changes things significantly.