The prompt is: Solve $au_x + bu_y = f(x,y)$, where $f(x,y)$ is a given function. If $a \neq 0$, write the solution in the form $u(x,y) = (a^2+b^2)^{-1/2}\int_L fds + g(bx-ay)$, where $g$ is an arbitrary function of one variable, L is the characteristic line segment from the y axis to the point (x,y), and the integral is a line integral. (Hint: Use the coordinate method.)
My attempt at a solution:
I changed variables to
$w = ax + by$
$e = bx - ay$
Now $ u_x = au_{w}+ bu_{e}$ and $u_y = bu_w-au_e$
so the PDE becomes $(a^2+b^2)u_w = f(x,y)$. Integrating both sides gives you $u = g(e) + \dfrac{\int{f}dw}{(a^2+b^2)}$
I am just having trouble getting from $\int{f}(a^2+b^2)^{-1}dw$ to $(a^2+b^2)^{-1/2}\int_L fds$
If I understand correctly, converting the integral to a line integral over the characteristic curve will introduce a factor of $(a^2+b^2)$, but I am new to PDE and I do not really know how to change the integral in $w$ to a line integral over the characteristic curve in $ds$.