I can't find any "tutorial" on how to solve this partial differential equation $$a\frac{\partial u}{\partial x}+b\frac{\partial u}{\partial y}+cu=0$$ I know how to solve equation in this form using the method of characteristics $$a\frac{\partial u}{\partial x}+b\frac{\partial u}{\partial y}=0$$ however the $cu$ term is really messing with me and I have no idea how to proceed, could anyone at least push me in the right direction?
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2Set $v=ue^{kx}$. This way, partial derivatives satisfy $v_y = u_ye^{kx}$ and $v_x = u_xe^{kx} + k v$, i.e $$av_x + bv_y = (au_x + bu_y + aku)e^{kx}.$$ Now, setting $k=c/a$ gives $av_x + bv_y =0$ which you already know how to solve. – EditPiAf Feb 27 '21 at 17:17