Consider the initial value problem: $$\pi U_x+U_y=0$$ $$U(x,0)-x^2$$
By solving with method of characteristics we find that: $$y'=\frac{1}{\pi} \implies y(x)=\frac{x}{\pi}+C \implies C=y-\frac{x}{\pi}$$ Because $U_x(x,y(x)) =0$ we have that: $$U(x,y)=f(C)=f(y -\frac{x}{\pi}) $$ and using our initial condition we can conclude that $U(x,y)=(y-\frac{x}{\pi})^2 $.
But here is where I have an issue, the solution given in my lecture notes uses the change of coordinates $$\bar{x}=\pi x+y $$
$$\bar{y}=x-\pi y $$
Because this PDE has constant coefficients we simply have that $U(x,y) = f(\bar{y})=f(x-\pi y)$.
But if we use our initial condition, we find that $$U(x,y)=(x- \pi y)^2 \neq (y-\frac{x}{\pi})^2 $$
Why is it that my method is incorrect? I can see that we can multiply $(y-\frac{x}{\pi})$ by $-\pi$ to get the same form as my lecturers solution. Are solutions equivalent up to a constant multiple?