Using the method of characteristics, show that the semilinear PDE
$\displaystyle (x+y)u_x + (x-y)u_y = 2x$
has the general solution $\displaystyle H(u-x-y, y^2 + 2xy -x^2) = 0
$
Find the solution for the case $\displaystyle u(x,0) = 2x $
Using the method of characteristics, show that the semilinear PDE
$\displaystyle (x+y)u_x + (x-y)u_y = 2x$
has the general solution $\displaystyle H(u-x-y, y^2 + 2xy -x^2) = 0
$
Find the solution for the case $\displaystyle u(x,0) = 2x $
It's homogeneous: $\displaystyle (x+y)dy=(x-y)dx$ so let $\displaystyle y=vx$. Gets a little messy but I get $\displaystyle y^2+2xy-x^2=h$. Still though I think we could let $\displaystyle w=y^2+2xy-x^2$ and $\displaystyle z=y$ to solve the PDE. I'm probably not doing it the way you want but I get for one solution:
$\displaystyle u(x,y)=2\left(y+1/2\sqrt{y^2-2xy+x^2}\right)+f(y^2+2xy-y^2)$ and supplying your side condition, then $\displaystyle f(x,y)=0$