<s>[INST] Stuck solving this PDE problem I need to solve $$uu_x+yu_y=x, u(x,1)=2x$$ i tried with characteristics method but got stuck solving the ODEs $$\frac{dx}{dt}=u ~,~\frac{dy}{dt}=y~,~\frac{du}{dt}=x$$ with initial values $x(0)=s,y(0)=1,u(0)=2s$. From the second one i get $y=e^t$, but how to solve the first two ODEs? I tried $$\frac{dx}{du}=\frac{u}{x} \implies x^2-u^2=k_1$$ where $k_1=-3s^2$ which gives $u=\sqrt{x^2+3s^2}$. Similarly $$\frac{dy}{y}=\frac{dx}{\sqrt{x^2+3s^2}} \implies y=k_2(x+\sqrt{x^2+3s^2})$$ And $k_2$ can be found out as $k_2=\frac{1}{3s}$, giving me $y=\frac{x+\sqrt{x^2+3s^2}}{3s}$. I'm not sure how to proceed further and solve for $u$?
 [/INST] According to your results we have
$$
\cases{
u^2-x^2-3s^2=0\\
y-\frac{x+\sqrt{x^2+3s^2}}{3s}=0
}
$$
or
$$
\cases{
u^2-x^2-3s^2=0\\
(3sy-x)^2-x^2-3s^2=0
}
$$
So solving the first for $s$ and substituting after choosing the feasible one into the second, we have
$$
u = x\frac{3y^2+1}{3y^2-1}
$$
</s>