# Coordinate method for solving first order linear PDE with non constant coefficient

Some background The so-called change of coordinates method (see Strauss) says that if $a,b$ are two constants such that $a^2 + b^2>0$ then

$a u_x + b u_y = 0$

is equivalent to

$w_\xi = 0$

provided that

$u(x,y) = w (\xi(x,y),\eta(x,y))$ and $\begin{cases} \xi(x,y) = a x + b y\\ \eta(x,y) = b x - a y \end{cases}$

Task I am trying to see if there is a chance that this method works for a first order linear PDE of the form

$a(x,y) u_x + b(x,y) u_y = 0$

where $a$ and $b$ are functions of $(x,y)$ which satisfy $a^2 + b^2 >0$.

Some calculations show that if $\begin{cases} a \xi_x + b \xi_y = \alpha(\xi,\eta)\\ a \eta_x + b \eta_y = \beta(\xi,\eta) \end{cases}$

then

$a u_x + b u_y = 0$

is equivalent to

$\alpha w_\xi + \beta w_\eta = 0.$

Question Except the elementary case where $a,b$ are constants (giving $\alpha = a^2 + b^2$ and $\beta = 0$), is there any interesting example ($a,b$ are not constant) where this method can be applied? The condition on $\xi$ and $\eta$ above seems very restrictive.

Thanks for help. m.