I'm trying to use the similarity method/dilation to solve the linear transport equation
$u_t+cu_x=0$. I make the following coordinate transformation: $(x,t)\mapsto (z,s)$ I apply the transformation $z=\epsilon^ax$ and $s=\epsilon^bt$.
The PDE becames $\epsilon^{b-c}v_s-k\epsilon^{a-c}v_z$=0
I think this is invariant when $c=a-b$.
I am unsure if the above steps are correct, or how to derive the ODE associated with this transformed PDE. I know once I solve the ODE I am basically done.