Task
I have the following equation: $$x^2u_{xx}-2u_{xy}+u_{yy}=0$$ which I need to reduce to its canonical form in the region where it is hyperbolic.
My solution
The region is the following: $x\in(-1,1), y\in \mathbb{R}$. The characteristic equations are: $$\dfrac{dx}{dy}=\dfrac{b\pm\sqrt{b^2-ac}}{c}=-1\pm\sqrt{1-x^2}$$
I found the following canonical variables to be (calculations were quite tedious, so I've done them in Wolfram): $$\xi=y - \dfrac{1}{x} + \dfrac{\sqrt{1 - x^2}}{x} + \arcsin x$$ $$\eta=y - \dfrac{1}{x} - \dfrac{\sqrt{1 - x^2}}{x} - \arcsin x$$ Substituting into the equation, get: $$u_{\xi\eta}=-\dfrac{x \left(-x^2+2 \sqrt{1-x^2}+2\right)}{2\left(1-x^2\right)^{3/2}}u_\xi -\dfrac{x \left(x^2+2 \sqrt{1-x^2}-2\right)}{2\left(1-x^2\right)^{3/2}}u_\eta$$
Questions
- As far as I understand, now I should express $x$ as a combination of $\xi$ and $\eta$ and substitute it into the right hand side. But is it possible to do with the given dependencies $\xi(x,y), \eta(x,y)$? If no, how one should handle such situations?
- Line $x=0$ is inside the region of interest. But, as far as I see, $\xi$ and $\eta$ are not defined at that line. Does this mean, that I should treat the case of $x=0$ separately considering the equation $0^2\cdot u_{xx}-2u_{xy}+u_{yy} = -2u_{xy}+u_{yy}=0$ and reducing it to the different canonical form? Or I can just plug $x=0$ into the previous solution $$u_{\xi\eta}=-\dfrac{x \left(-x^2+2 \sqrt{1-x^2}+2\right)}{2\left(1-x^2\right)^{3/2}}u_\xi -\dfrac{x \left(x^2+2 \sqrt{1-x^2}-2\right)}{2\left(1-x^2\right)^{3/2}}u_\eta$$ since those coeffitients at right hand side behave well on this line?