Change of variable in two variables differential equation I have a problem in understanding a passage from the nots of a professor of us.
The starting problem is this PDE:
$$
\dfrac{\partial^2 t}{\partial u^2} = \frac{1}{h}\frac{\partial}{\partial h}\left(h^2 \frac{\partial t}{\partial h}\right)$$
Where $ t = t(u, h)$.
Then he says that by the following change of variable
$$
\begin{cases}
\xi = 2\sqrt{h} - u \\\\
\eta = 2\sqrt{h} + u
\end{cases}
$$
We should get
$$\frac{\partial^2 t}{\partial\xi\partial\eta} = 0$$
Simply.
Yet I have done the calculations five times, and what I end up with is:
$$\frac{\partial^2 t}{\partial\xi\partial\eta} = \frac{-2}{\eta + \xi}\left(\frac{\partial t}{\partial \xi} + \frac{\partial t}{\partial\eta}\right)$$
I am not asking anyone of you to do the complete calculation, for I know it's so tedious... But I really cannot get out of this. Any hint? 
It could also be that he wrote wrong? Or the change of variable is not correct? 
Thank you in advance!
 A: Note that the canonical form may include first-order terms as well. Using the proposed change of variable, we compute the partial derivatives by using the chain rule:
\begin{aligned}
t_u &= t_\xi\xi_u + t_\eta\eta_u = -t_\xi + t_\eta \\
t_h &= t_\xi\xi_h + t_\eta\eta_h = (t_\xi + t_\eta) h^{-1/2} .
\end{aligned}
The product rule gives $h^{-1}(h^2t_h)_h = 2t_h + ht_{hh} $. Similarly to above, the second derivatives satisfy
\begin{aligned}
t_{uu} &= -t_{u\xi} + t_{u\eta} = t_{\xi\xi} - 2t_{\xi\eta} + t_{\eta\eta} \\
t_{hh} &= (t_{h\xi} + t_{h\eta}) h^{-1/2} - \tfrac12 (t_\xi + t_\eta) h^{-3/2} \\
 &= (t_{\xi\xi} + 2t_{\xi\eta} + t_{\eta\eta})/h - \tfrac12 (t_\xi + t_\eta) h^{-3/2} .
\end{aligned}
Using $\frac1{4}(\xi + \eta) = \sqrt{h}$, we are now left with
$$
t_{\xi\eta} - \tfrac32 (\xi+\eta)^{-1} (t_\xi + t_\eta) = 0\, ,
$$
which doesn't yield $t_{\xi\eta} = 0$. With this 2nd-order hyperbolic PDE ($h>0$), integration of
$\frac{\text dh}{\text du} = \pm\sqrt{h}$ leads indeed to the characteristic variables $\xi= \pm 2\sqrt{h} - u$ and $\eta = \pm2\sqrt{h} + u$.
Please let me know if you spot any mistake.
