A general solution of the pde $UU_x + UU_y = x$ is of the form $f(u^2 -x^2 , {y \over (x+u)}) = 0 $ where $f : \Bbb R^2 \to \Bbb R$ is $C^1$ and $\nabla f \neq 0$ for every point.
Can anyone help me out to solve this problem? I know the answer would have been correct if the part "$\nabla f \neq 0$ for every point" was not there. I really do not know what does it mean? and what is the relation with this problem?
Thank You in advance.
Using the characteristics method
$$ \frac{dx}{1}=\frac{dy}{1}=\frac{udu}{x} $$
then
$$ \frac{dx}{1}=\frac{dy}{1}\Rightarrow y-x=C_1 $$
as well as
$$ \frac{dx}{1}=\frac{udu}{x}\Rightarrow u^2-x^2 = C_2 $$
and finally
$$ f(x,y,u)=u^2-x^2 - \phi(y-x)=0 $$