The problem statement is to simplify it, i.e. give it a canonical form to this differential equation using the characteristics to achieve that. However, even after many hints and trials on similar problems, I am still unable to proceed further with this one.
$$u_{xx}-2u_{xy}+u_{yy}+9u_x+9u_y-9u=0.$$
Here is all the additional hints that I've been given by my professor:
We need to write the characteristic equation $A dy = (B \pm \sqrt{B^2-AC}) dx$, integrate to solve it, write parameters $\lambda$ and $\mu$, $y=(1\pm …)x+\text{const}$ and \begin{cases} \lambda= y-(1+\cdots)x\\\mu=y-(1-…)x\end{cases}
Then we basically what we are left to do is to turn $u(x,y)$ into $u(\lambda(x,y), \mu(x,y))$ by calculating $\frac{\partial u}{\partial x}=\frac{\partial u}{\partial \lambda}\cdot\frac{\partial \lambda}{\partial x}+\frac{\partial u}{\partial \mu}\cdot\frac{\partial \mu}{\partial x}$ and etc.
Despite being provided all the hints, I am still struggling to connect the dots between individual hints (maybe because of some persisent unidentified error in my calculations). I would appreciate if someone could provide a solution to this problem.
Best Answer
Note that
$$ u_{xx}-2u_{xy}+u_{yy}+9u_x+9u_y-9u=\left(\left(\partial_x-\partial_y\right)^2+9\left(\partial_x+\partial_y\right)-9\right)u=\left(\left(\partial_{x-y}\right)^2+9\left(\partial_{x+y}\right)-9\right)u=0 $$ if we change variables to
$$ \cases{ \eta=x-y\\ \xi=x+y } $$
$$ \left(\left(\partial_{\eta}\right)^2+9\left(\partial_{\xi}\right)-9\right)u\left(\frac{\xi+\eta}{2},\frac{\xi-\eta}{2}\right)=\left(\left(2\partial_{\eta}\right)^2+9\left(2\partial_{\xi}\right)-9\right)u\left(\xi+\eta,\xi-\eta\right)=0 $$
or
$$ 4u_{\eta\eta}+18u_{\xi}-9u = 0 $$