I am asked to solve a PDE in the sense of distributions, but I'm kinda confused by the definition. So I am trying to solve the PDE: $$au_x+bu_y=0$$ Now, let us define $f(bx-ay) = u(x,y)$, where $f$ is only locally integrable. I'm supposed to show that $u(x,y)$ solves this PDE in the sense of distributions. The hint is to make a change of variable so the characteristic directions are the coordinate axes.
Now, I began to solve it: $$\left< au_x+bu_y ,\phi \right> = \iint_{\mathbb R^2} -u(a\phi_x+b\phi_y) dx dy$$ Where $\phi\in C^{\infty}_c (\mathbb R^2)$. But how do I proceed from here? I'm kinda confused by what the question is asking.