# Solving the PDE: $y u_x + x u_y = 1, u(0,y) = e^{-y^2}$

I'm trying to solve this PDE $yu_x + xu_y = 1$, $u(0,y) = e^{-y^2}$ with the method of characteristics, but I do not have a great understanding of the technique. I know what's going on - the characteristics being the curves that $u(x,y)$ travels along in the $x$-$y$ plane where $u$ is unchanged. But I can't follow the method to solve this problem.

I think $\frac{dx}{dt} = y,\ \frac{dy}{dt} = x$ so then $x(t) = c_1\cos(t)+c_2\sin(t)$ and $y(t) = c_3\cos(t) + c_4\sin(t)$. But beyond that I have no idea what's going on.