Try let $v=x+y$ , $w=x-y$ ,
Then $\dfrac{\partial u}{\partial x}=\dfrac{\partial u}{\partial w}\dfrac{\partial w}{\partial x}=\dfrac{\partial u}{\partial w}$
$\dfrac{\partial u}{\partial y}=\dfrac{\partial u}{\partial w}\dfrac{\partial w}{\partial y}=-\dfrac{\partial u}{\partial w}$
$\dfrac{\partial^2u}{\partial y^2}=\dfrac{\partial}{\partial y}\left(-\dfrac{\partial u}{\partial w}\right)=-\dfrac{\partial}{\partial v}\left(\dfrac{\partial u}{\partial w}\right)\dfrac{\partial v}{\partial y}=-\dfrac{\partial^2u}{\partial vw}$
$\therefore-\dfrac{\partial^2u}{\partial vw}=\dfrac{\partial u}{\partial w}$
Let $z=\dfrac{\partial u}{\partial w}$ ,
Then $\dfrac{\partial z}{\partial v}=\dfrac{\partial^2u}{\partial vw}$
$\therefore-\dfrac{\partial z}{\partial v}=z$
$\dfrac{dz}{z}=-~dv$
$\int\dfrac{dz}{z}=\int-~dv$
$\ln z=-v+c_1(w)$
$z=c_2(w)e^{-v}$
$\dfrac{\partial u}{\partial w}=c_2(w)e^{-v}$
$u=\int c_2(w)e^{-v}~dw$
$u=C_1(v)+C_2(w)e^{-v}$
$u=C_1(x+y)+C_2(x-y)e^{-x-y}$
I did it correctly in Finding an analytical solution to the wave equation using method of characteristics , and did it not known whether correct or not in Solving $yu_{xx}+(x+y)u_{xy}+xu_{yy}=0$ , but why in here is wrong?