Find the solutions of the PDE $$ 2v_{\xi\eta}+v_{\eta}=0.~~~~~(+) $$
Hello, with this question I refer to Transforming hyperbolic PDE into normal form. I sum it up what happened there:
The original PDE was $$ x^2u_{xx}-y^2u_{yy}=0~~~~\text{in}~~~~\Omega:=\left\{(x,y)\in\mathbb{R}^2: x>0,y>0\right\}~~~~~(++). $$ Using the transformation $$ \xi:=\ln(y/x),~~~~~\eta:=-\ln(xy), $$ the PDE (++) was transformed into the PDE (+).
Now to find the solutions of (++), it is easier to find the solutions of (+). And that's what I want to do now:
I use $$ v_{\xi\eta}=v_{\eta\xi} $$ to write $$ 2v_{\xi\eta}+v_{\xi}=2\cdot (v_{\xi})_{\eta}+v_{\xi}=0 $$ and then I substitute $z:=v_{\xi}$, getting the ODE $$ z_{\eta}=-\frac{1}{2}z. $$ Using "separation of the variables", I get $$ z=\exp\left(-\frac{1}{2}\eta+C_1(\xi)\right). $$ Re-substituting, that means $$ v_{\xi}=\exp\left(-\frac{1}{2}\eta+C_1(\xi)\right) $$ and integrating to $\xi$ gives me $$ v(\xi,\eta)=\int_{\xi_0}^{\xi}\exp\left(-\frac{1}{2}\eta+C_1(\tau)\right)\, d\tau+C_2(\eta), $$ which means for the original PDE (++) in my opinion, that the solutions are given by $$ u(x,y)=\int_{z_0}^{\ln(y/x)}\exp\left(\frac{1}{2}\ln(xy)+C_1(\tau)\right)\, d\tau+C_2(-\ln(xy)). $$
Can anybody tell me if this is right, i.e. if this are the right solutions?
With kind regards and thank for your help!
